Docstoc

ENGR201 Lecture 13 — Sampling

Document Sample
ENGR201 Lecture 13 — Sampling Powered By Docstoc
					Overview       Ideal Sampling   Reconstruction    Practical Sampling 1: Pulse-Train Sampling   Practical Sampling 2: Zero-Order Hold   Summary




                                                  ENGR201
                                           Lecture 13 — Sampling

                                                     Pawel Dmochowski
                                                 pawel.dmochowski@vuw.ac.nz



                                                        Wed, 9 Sept 2009




ENGR201, Lecture 13 — Sampling                                                                                          Wed, 9 Sept 2009   1/29

Overview       Ideal Sampling   Reconstruction    Practical Sampling 1: Pulse-Train Sampling   Practical Sampling 2: Zero-Order Hold   Summary




           1   Overview

           2   Ideal Sampling

           3   Reconstruction

           4   Practical Sampling 1: Pulse-Train Sampling

           5   Practical Sampling 2: Zero-Order Hold

           6   Summary




ENGR201, Lecture 13 — Sampling                                                                                          Wed, 9 Sept 2009   2/29
        Overview         Ideal Sampling           Reconstruction   Practical Sampling 1: Pulse-Train Sampling                   Practical Sampling 2: Zero-Order Hold     Summary



        Digitization Overview


                              So far we have focused mainly on continuous-time signals - they were
                              defined (or observed) for any value of t
                              In addition, these signal were also assumed to take on an arbitrary value
                              Since most signal processing is typically carried out on a digital computer,
                              we need a way to represent a continuous signal digitally
                              The process of analog-to-digital conversion (ADC) requires us to discritize
                              both the time and amplitude. This is accomplished in two steps
                                    1     sampling - the conversion of a signal from continuous to discrete in
                                          time
                                    2     quantization - the conversion of the signal samples from continuous
                                          to discrete in amplitude




       ENGR201, Lecture 13 — Sampling                                                                                                                       Wed, 9 Sept 2009   3/29

         Overview     Ideal Sampling    Reconstruction             Practical Sampling 1: Pulse-Train Sampling                   Practical Sampling 2: Zero-Order Hold     Summary
P1: RPU/XXX     P2: RPU/XXX     QC: RPU/XXX     T1: RPU
CUUK852-Mandal & Asif         May 25, 2007        19:8
        Digitization Overview
               396            Sampling is aIIIprocess of observing/capturing the signal at regular
                                         Part  Discrete-time signals and systems

                              intervals.
               Fig. 9.2. Time-domain                                       x(t)                                            xs(t) with Ts = T
               illustration of sampling as a
               product of the band-limited
               signal and an impulse train.
                                                                                              t                                                         t
               (a) Original signal x(t );                              0                            −6T −4T −2T        0        2T   4T       6T
               (b) sampled signal x s (t ) with
               sampling interval T s = T ;          (a)                                           (b)
                              This operation, if done properly, is lossless - we can recover (reconstruct)
               (c) sampled signal x s (t ) with
                                                                                x (t) with T = 2T                           s             s
                              the original signal with no loss of information
               sampling interval T s = 2T .


                              This lecture covers the theory of sampling and reconstruction - introduces
                                                                                          t
                                                                   −6T −4T −2T 0 2T 4T 6T
                              Nyquist Sampling Theory
                                                                                                  (c)



                                                                     which signal amplitude is signal
                              Quantization is a process in Eq. (9.4), the CTFT X s (ω) of the sampledassigned to a level
                                             Calculating the CTFT of
                                         xs (t) is given values. Each of the allowed levels can then represented
                              from a finite set of by
                              in binary.                     ∞
                                                                           1             ∞
                                                    X s (ω) = ℑ x(t)              δ(t − kTs ) =         F{x(t)} ∗ ℑ             δ(t − kTs )
                                                                  2π
                              The process of quantization is lossy - restricting the amplitude to a finite
                                                     k=−∞                    k=−∞
                                                             ∞
                              number of values1is X (ω) approximation.
                                                  an ∗ 2π           2mπ    1 ∞         2mπ
                                            =                  δ ω−      =        X ω−
                                                              2π                  Ts   m=−∞             Ts        Ts   m=−∞                        Ts
                              Next lecture will discuss various methods of quantization, and will focus
                                                                                      (9.5)
                              mainly on how to minimize the loss of fidelity
                                                    where ∗ denotes the CT convolution operator. In deriving Eq. (9.5), we used
       ENGR201, Lecture 13 — Sampling the following CTFT pair:                                                                                              Wed, 9 Sept 2009   4/29
Overview       Ideal Sampling   Reconstruction    Practical Sampling 1: Pulse-Train Sampling   Practical Sampling 2: Zero-Order Hold   Summary



Preliminaries


                  f (t)δ(t − t0 ) = f (t0 )δ(t − t0 )



                                                 ∞
                  sifting property:              −∞
                                                      f (t)δ(t − t0 ) = f (t0 )



                                                                                    ∞
                  convolution: f (t) ∗ g(t) = g(t) ∗ f (t) =                        −∞
                                                                                           f (τ )g(t − τ )dτ



                  f (t) ∗ δ(t − t0 ) = f (t − t0 )




ENGR201, Lecture 13 — Sampling                                                                                          Wed, 9 Sept 2009   5/29

Overview       Ideal Sampling   Reconstruction    Practical Sampling 1: Pulse-Train Sampling   Practical Sampling 2: Zero-Order Hold   Summary




           1   Overview

           2   Ideal Sampling

           3   Reconstruction

           4   Practical Sampling 1: Pulse-Train Sampling

           5   Practical Sampling 2: Zero-Order Hold

           6   Summary




ENGR201, Lecture 13 — Sampling                                                                                          Wed, 9 Sept 2009   6/29
          Overview     Ideal Sampling    Reconstruction          Practical Sampling 1: Pulse-Train Sampling   Practical Sampling 2: Zero-Order Hold   Summary



          The Sampling Theorem

                          Consider a continuous signal x(t), whose FT satisfies a condition

                                                                      X(ω) = 0                for |ω| > 2πβ


                          x(t) is said to be bandlimited to 2πβ radians/s or β Hz.
                          We will next show a key result of sampling theory :

                  Theorem (Nyquist Sampling Theorem)
                  A signal x(t) bandlimited to β Hz can be transformed to a discrete-time
                  sequence of samples xk with no loss of information provided that the sampling
                  frequency (or rate) satisfies fs > 2β. The frequency fs is referred to as the
                  Nyquist frequency

                          In other words, the sampling interval Ts must satisfy Ts < 1/2β.



         ENGR201, Lecture 13 — Sampling                                                                                                Wed, 9 Sept 2009   7/29

          Overview     Ideal Sampling    Reconstruction          Practical Sampling 1: Pulse-Train Sampling   Practical Sampling 2: Zero-Order Hold   Summary



          The Sampling Theorem: Proof
                          Let us prove Nyquist Sampling Theorem
                          The discrete time version of the signal x(t) is obtained by multiplication
                          by an impulse train
                                                                                          ∞
                                                                        s(t) =                  δ(t − kTs )
                                                                                    k=−∞
                   where Ts denotes the sampling interval, that is the spacing between
                   adjacent impulses.
RPU/XXX     QC: RPU/XXX    T1: RPU
sif  May 25, 2007  A related parameter is the sampling rate fs = 1/Ts = ωs /2π
                    19:8
                   Mathematically, the sampled version of x(t) is given by xs (t) = x(t)s(t),
                   that is
                                                                     ∞                                 ∞
                                       xs (t) = x(t)              δ(t −
                              Part III Discrete-time signals and systems kTs )                  =             x(kTs )δ(t − kTs )
                                                                  k=−∞                              k=−∞


 2. Time-domain                                           x(t)                                                     xs(t) with Ts = T
ation of sampling as a
ct of the band-limited
  and an impulse train.
                                                                                      t                                                   t
 iginal signal x(t );                                 0                                       −6T −4T −2T      0     2T    4T     6T
mpled signal x s (t ) with
           ENGR201, ;            (a)
ing interval T s = T Lecture 13 — Sampling                                                (b)                                          Wed, 9 Sept 2009   8/29
           CUUK852-Mandal & Asif          May 25, 2007         19:8
Overview     Ideal Sampling     Reconstruction   Practical Sampling 1: Pulse-Train Sampling         Practical Sampling 2: Zero-Order Hold           Summary



The Sampling Theorem: Proof
                              396                               Part III Discrete-time signals and systems
                The FT of xs (t), denoted by Xs (ω) is
                                                ∞                                  ∞
                       Fig. 9.2. Time-domain                      1 x(t)     2π               2mπs(t) with Ts = T
                                                                                                  x
                Xs (ω) = F of sampling as a δ(t − kTs ) =
                       illustration x(t)                             X(ω) ∗           δ ω−
                       product of the band-limited               2π          Ts m=−∞           Ts
                                          k=−∞
                       signal and an impulse train.
                where we Original signal x(tconvolution property and the FT pair t −6T −4T −2T 0 2T 4T 6T
                       (a) used the );                             0
                       (b) sampled signal x s (t ) with
                                      ∞                          ∞
                       sampling interval T s = T ;      (a) 2π                2mπ (b)
                                                δ(t − kTs )
                       (c) sampled signal x s (t ) with
                                                                         δ ω−
                                                            Ts m=−∞             Ts                xs(t) with Ts = 2T
                                    k=−∞
                       sampling interval T s = 2T .
                                                                              2π
                Finally we can show that, for ωs =                            Ts
                                                                              ∞                                         −6T −4T −2T             0       2T      4T   6T
                                                          1
                                                 Xs (ω) =                            X (ω − mωs )                     (c)
                                                          Ts           m=−∞

                the last step uses the fact that
                                                           ∞        Calculating the CTFT of Eq. (9.4), the CTFT X s (ω) of the sampled si
                                                                                     ∞
                                           f (t) ∗              xs (t) − given by
                                                                δ(t    is nT ) =         f (t − nT )
                                                      k=−∞                               k=−∞
                                                                                          ∞                                                     ∞
                                                                                                     1
                                                                X s (ω) = ℑ x(t)      δ(t − kTs ) =    F{x(t)} ∗ ℑ                                      δ(t − kTs )
                (prove this for homework!,                     using the definition of convolution and the
                                                                                 k=−∞
                                                                                                    2π                                      k=−∞
                sifting property of δ(ω)                                       1               2π      ∞
                                                                                                                        2mπ         1 ∞            2m
                                                                          =          X (ω) ∗                δ ω−                =             X ω−
ENGR201, Lecture 13 — Sampling                                                2π               Ts   m=−∞                            Ts 2009
                                                                                                                         Ts Wed, 9 Sept m=−∞ 9/29   T
Overview     Ideal Sampling     Reconstruction   Practical Sampling 1: Pulse-Train Sampling         Practical Sampling 2: Zero-Order Hold           Summary



The Sampling Theorem: Proof the CT convolution operator. In deriving Eq. (9.5), we
                      where ∗ denotes
                                                                the following CTFT pair:
                                                                                         ∞                                  ∞
                                                                                               2π          CTFT                             2mπ
                We have shown that xs (t) has a                                    δ(t − kT )
                                                                       Fourier transform sof←→ Ts                                 δ ω−
                        Fig. 9.3. Frequency-domain                            k=−∞                   m=−∞                                    Ts
                        illustration of the impulse-train            ∞
                        sampling. (a) Spectrum X(ω) of        1
                                                          based on entry (19) of Table 5.2. Equation (9.5) implies that the spectrum X
                        the original signal x(t );
                                                  Xs (ω) =                  X (ω − mωs )
                                                          of Ts sampled signal xs (t) is a periodic extension, consisting of the sh
                                                              the
                        (b) spectrum Xs (ω) of the                m=−∞
                                                          replicas of the spectrum X (ω) of the original baseband signal x(t). Figure
                        sampled signal xs (t ) with
                        sampling rate ωs ≥ 4πβ; (c)
                                                          illustrates the frequency-domain interpretation of Eq. (9.5). The spectrum o
                That is,spectrumprocess sampled original signal x(t)aiscontinuous-time signaltrapezoidal waveform a
                           the Xs (ω) of the of uniformly sampling                assumed to be an arbitrary results
                                                           spectrum 9.3(a). The spectrum X s (ω) to the
                in a periodic(t extentionrate its shown in Fig. with the period equal of the sampled signal xs (t) is plo
                        signal xs ) with sampling of
                        ωs < 4πβ.
                sampling frequency
                                                 X(w)                                             Xs(w) with ws ≥ 4pb                                Xs(w) with ws < 4
                                                 1                                             1/Ts

                                                                      w                                                 w
                                        −2pb 0       2pb                           −ws   −2pb 0     2pb      ws                    −2ws −ws         0     ws      2ws
                                                                                                                                   (ws − 2pb)                    2pb

                For a signal bandlimited to 2πβ, the replicas X(ω − mωs ) do not overlap
                        (a)                        (b)                       (c)
                provided that ωs ≥ 4πβ.
                In the case of non-overlapping spectal components, we can recover the
                original X(ω) (and thus x(t)) by low-pass filtering the spectrum Xs (ω).

ENGR201, Lecture 13 — Sampling                                                                                                  Wed, 9 Sept 2009        10/29
        Overview     Ideal Sampling        Reconstruction   Practical Sampling 1: Pulse-Train Sampling    Practical Sampling 2: Zero-Order Hold   Summary
entry (19) of Table 5.2. Equation (9.5) implies that the spectrum X s (ω)
         signal xs (t) is a periodic extension, Proof
mpled The Sampling Theorem: consisting of the shifted
 f the spectrum X (ω) of the original baseband signal x(t). Figure 9.3
  the frequency-domain interpretation of Eq. (9.5). The spectrum of the
ignal x(t) is assumed to be an arbitrary trapezoidal waveform and is
Fig. 9.3(a). The spectrum X s (ω) of the sampled signal xs (t) is plotted
                        If ωs < 4πβ, the specral components will overlap, resulting in aliasing
                      Xs(w) with ws ≥ 4pb                                        Xs(w) with ws < 4pb
                     1/Ts                                                                                1/Ts

                                                   w                                                       w
      −ws   −2pb 0       2pb          ws                     −2ws −ws           0     ws     2ws
                                                             (ws − 2pb)                     2pb

(b)                     In the case aliased(c) s (ω), cannot perfectly recover the original spectrum
                                            X
                        X(ω) by means of lowpass filtering
                                             ˆ
                        The recovered signal x(t) will contain distortion




        ENGR201, Lecture 13 — Sampling                                                                                            Wed, 9 Sept 2009   11/29

        Overview     Ideal Sampling        Reconstruction   Practical Sampling 1: Pulse-Train Sampling    Practical Sampling 2: Zero-Order Hold   Summary



        Aliasing

                        In practice, information-bearing signals are not strictly bandlimited to β,
                        there will exist some spectral component beyond this frequency
                        Thus we need to introduce corrective measures for aliasing, such as:
                            1    before sampling, pass the signal through a low-pass pre-alias filter to
                                 attenuate the high-frequency components not essential to the
                                 information being carried
                            2    sample the signal at a higher rate than 1/2β - this is referred to as
                                 oversampling ; this increases the spacing between the replicas of the
                                 spectrum
                        Note also, that in order to recover a signal sampled at precisely the
                        Nyquist frequency, we need an ideal lowpass filter - this is not realistic.
                        Thus, in practice, the reconstruction of the original signal cannot be
                        accomplished unless the signal is oversampled, allowing for more relaxed
                        (realistic) specifications for the reconstruction filter



        ENGR201, Lecture 13 — Sampling                                                                                            Wed, 9 Sept 2009   12/29
        Overview         Ideal Sampling      Reconstruction        Practical Sampling 1: Pulse-Train Sampling       Practical Sampling 2: Zero-Order Hold      Summary




: RPU/XXX      P2: RPU/XXX                QC: RPU/XXX             T1: RPU
UUK852-Mandal & Asif           May 25, 2007         19:8


                   1     Overview

              398                                      Part III Discrete-time signals and systems
               2         Ideal Sampling

                     Reconstruction of
              9.1.1 Reconstruction a band-limited signal from its samples
               3
                                                Figure 9.3(b) illustrates that the CTFT X s (ω) of the sampled signal xs (t) is a
                                                periodic extension of the CTFT of the original signal x(t). By eliminating the
                   4     Practical                        Pulse-Train Sampling
                                           Sampling 1: in X s (ω), we should be able to reconstruct x(t). This is accomplished
                                                replicas
                                                by applying the sampled signal xs (t) to the input of an ideal lowpass filter (LPF)
                   5     Practical              with 2: Zero-Order function:
                                           Sampling the following transferHold
                                                                                                       Ts       |ω| ≤ ωs /2
                                                                                            H (ω) =                                                         (9.7)
                   6     Summary                                                                       0        elsewhere.

                                              The CTFT Y (ω) of the output y(t) of the LPF is given by Y (ω) = X s (ω)H (ω),
                                              and therefore all shifted replicas at frequencies ω > ωs /2 are eliminated. All
                                              frequency components within the pass band ω ≤ ωs /2 of the LPF are amplified
                                              by a factor of Ts to compensate for the attenuation of 1/Ts introduced during
                                              sampling. The process of reconstructing x(t) from its samples in the frequency
                                              domain is illustrated in Fig. 9.4. We now proceed to analyze the reconstruction
       ENGR201, Lecture 13 — Sampling         process in the time domain.                                          Wed, 9 Sept 2009 13/29

        Overview    Ideal Sampling                                                               multiplication in the frequency
                                                  According to the convolution property,Practical Sampling 2: Zero-Order Hold
                                     Reconstruction   Practical Sampling 1: Pulse-Train Sampling                                   Summary
                                              domain transforms to convolution in the time domain. The output y(t) of
        Reconstruction                        the lowpass filter is therefore the convolution of its impulse response h(t)
                                              with the sampled signal xs (t). Based on entry (17) of Table 5.2, the impulse
                                              response of an ideal lowpass filter with the transfer function given in Eq. (9.7) is
                        The frequency given by representation of the sampled signal has hinted at
                                               domain
                             the process of reconstruction of the original x(t)
                                                                               s                                    ωt
                             Let us now examine it in more detail h(t) = sinc 2π .                                                                          (9.8)

                                                                         x(t), can be recovered by signal, x (t)
                             The original signal X(ω), and thus response h(t) with the sampledeliminating =
                                             Convolving the impulse                                         s
                             the spectral replicas in Xs (ω) by applying an ideal lowpass filter (LPF)
                                               ∞

                             with a transfer k=−∞ x(kTs )δ(t − kTs ) yields
                                              function
              Fig. 9.4. Reconstruction of the                                             ∞
              original baseband signal x(t ) by                              Ts ωs t ∗
                                                                    y(t) = sinc |ω| ≤ ωs /2 s )δ(t − kTs ),
                                                                                             x(kT                                                           (9.9)
                                                              H(ω) =             2π
              ideal lowpass filtering.                                        0          k=−∞
                                                                                  elsewhere
              (a) Spectrum of the sampled
              signal xs (t ); (b) transfer function which reduces to
                            The FT
              H(ω) of the lowpass filter;of the LPF output y(t) is Y (ω) = Xs (ω)H(ω), and thus                                                   all
              (c) spectrum of the                                           ∞
                            shifted replicas at frequencies ω > ωs /2 are suppressed.       ωs t
              reconstructed signal x(t ).                          y(t) =      x(kTs ) sinc       ∗ δ(t − kTs )                                         (9.10)
                             All frequencies ω ≤ ωs /2 are amplified by Ts to 2π
                                                             k=−∞            compensate for the
                             attenuation of 1/Ts during the sampling process.
                                     Xs(w)                                                 H(w)                                          Y(w)
                                     1/Ts                                                 Ts                                             1

                                                              w                                                 w                                              w
                       −ws   −2pb 0 2pb           ws                         −ws /2   0      ws /2                              −2pb 0     2pb


       ENGR201, Lecture 13 — Sampling                                                                                                       Wed, 9 Sept 2009        14/29
         Overview      Ideal Sampling          Reconstruction           Practical Sampling 1: Pulse-Train Sampling                   Practical Sampling 2: Zero-Order Hold              Summary



         Reconstruction

                             Now, let’s analyse reconstruction in the time domain
                             The output of the reconstruction filter y(t) is the convolution of the
                             sampled sequence xs (t) and the reconstruction filter impulse response
                             h(t), given by
                                                                      ωs t
                                                        h(t) = sinc
                                                                      2π
                             Recalling that
                                                                              ∞                                           ∞
                                           xs (t) = x(t)                               δ(t − kTs ) =                               x(kTs )δ(t − kTs )
                                                                         k=−∞                                    k=−∞

                             the LPF output is given by
                                                                                                                            ∞
                                                                                                    ωs t
                                        y(t) = xs (t) ∗ h(t) = sinc                                              ∗                    x(kTs )δ(t − kTs )
                                                                                                    2π
                                                                                                                      k=−∞




       ENGR201, Lecture 13 — Sampling                                                                                                                                  Wed, 9 Sept 2009   15/29

         Overview      Ideal Sampling          Reconstruction           Practical Sampling 1: Pulse-Train Sampling                   Practical Sampling 2: Zero-Order Hold              Summary



         Reconstruction
                             This in turn reduces to
                                                                             ∞
                                                                                                                 ωs t
PU/XXX         P2: RPU/XXX       QC: RPU/XXX          y(t) =
                                                        T1: RPU                       x(kTs ) sinc                                 ∗ δ(t − kTs )
K852-Mandal & Asif         May 25, 2007         19:8
                                                                                                                 2π
                                                             k=−∞
     P1: RPU/XXX           P2: RPU/XXX           QC: RPU/XXX   T1: RPU
                                                                             ∞
     CUUK852-Mandal & Asif            May 25, 2007              19:8                                             ωs (t − kTs )
                                                                  =                   x(kTs ) sinc
                                                                                                                       2π
                                                              k=−∞
         399                                   9 Sampling and quantization
                             This implies that the original signal x(t) is reconstructed by adding a
                     399                      9 Sampling and quantization
                             series of time shifted sinc functions whose amplitudes are scaled by the
                                                                            h(t) sinc ( s )
                                   x (t)                                               wt
                             valuess of the samples at the center of the sinc= function2p

                                                   xs(t)                                                                                  h(t) = sinc   (w t )
                                                                                                                                                          s
                                                                                                                                                         2p
                                                                         t                                                                                               t
               −3Ts −2Ts −Ts      0       Ts       2Ts      3Ts                       −4Ts −3Ts −2Ts −Ts              0       Ts      2Ts      3Ts      4Ts
                                                                                        t                                                                                           t
         (a)               −3Ts −2Ts −Ts          0        Ts     2Ts        3Ts(b)               −4Ts −3Ts −2Ts −Ts                  0      Ts      2Ts         3Ts   4Ts

                     (a)                                                                    (b) y(t)

                                                                                                               y(t)
                                                                                                                                                                         t
                                                                −4Ts −3Ts −2Ts −Ts            0     Ts     2Ts        3Ts      4Ts
                                                                                                                                                                                    t
                            (c)
       ENGR201, Lecture 13 — Sampling                                         −4Ts −3Ts −2Ts −Ts           0      Ts          2Ts      3Ts     4Ts                     Wed, 9 Sept 2009   16/29
                                                                                                                    2p
        Overview       Ideal Sampling   Reconstruction   Practical Sampling 1: Pulse-Train Sampling     Practical Sampling 2: Zero-Order Hold     Summary



        Reconstruction                              t                                                                                                t
Ts −Ts        0        Ts       2Ts     3Ts                    −4Ts −3Ts −2Ts −Ts                 0         Ts     2Ts    3Ts     4Ts

                                                         (b)

                                                                               y(t)



                                                                                                                                                     t
                                         −4Ts −3Ts −2Ts −Ts                0      Ts       2Ts        3Ts    4Ts

           (c)              Note that at time t = kTs only the kth sinc (with amplitude x(kTs )) is
                            non-zero. The value of the reconstructed signal at t = kTs is thus x(kTs ).
onstruction of the          or
                            The values in between two samples are interpolated using a linear
signal in the time          combination of time-shifted sinc functions
                                                      ∞
ampled signal                                                                                    ωs (t − kTs )
                            The sinc functiony(t) =        x(kTs ) interpolating function
                                              is referred to as the sinc                .                                                       (9.11)
ulse response h(t )                                                                                   2π
                                                               k=−∞
s filter;
 ted signal x(t )         Equation (9.11) implies that the original signal x(t) is reconstructed by adding
onvolving xs (t )         a series of time-shifted sinc functions, whose amplitudes are scaled according
                          to the values of the samples at the center location of the sinc functions. The
                          sinc functions in Eq. (9.11) are called the interpolating functions and the over-
        ENGR201, Lecture 13 — Sampling                                                                                Wed, 9 Sept 2009 17/29

         Overview
                          all process is referred to as the 1:band-limited interpolation. The time-domain
                     Ideal Sampling   Reconstruction Practical Sampling Pulse-Train Sampling Practical Sampling 2: Zero-Order Hold    Summary
                          interpretation of the reconstruction of the original band-limited signal x(t) is
                          illustrated in Fig. 9.5. At t = kTs , only the kth sinc function, with amplitude
                          x(kTs ), is non-zero. The remaining sinc functions are all zero. The value of the
                          reconstructed signal at t = kTs is therefore given by x(kTs ). In other words,
                  1 Overview
                          the values of the reconstructed signal at the sampling instants are given by the
                          respective samples. The values in between two samples are interpolated using
                          a linear combination of the time-shifted sinc functions.
                  2 Ideal Sampling


                            Example 9.1
                   3   Reconstruction
                            Consider the following sinusoidal signal with the fundamental frequency f 0 of
                            4 kHz:
                   4   Practical Sampling 1: Pulse-Train Sampling
                                                         g(t) = 5 cos(2π f 0 t) = 5 cos(8000πt).
                   5   Practical Sampling 2: Zero-Order Hold
                         (i) The sinusoidal signal is sampled at a sampling rate f s of 6000 samples/s
                   6   Summary reconstructed with an ideal LPF with the following transfer function:
                             and
                                                                             1/6000 |ω| ≤ 6000π
                                                           H1 (ω) =
                                                                             0      elsewhere.
                                  Determine the reconstructed signal.

        ENGR201, Lecture 13 — Sampling                                                                                          Wed, 9 Sept 2009    18/29
          Overview          Ideal Sampling   Reconstruction        Practical Sampling 1: Pulse-Train Sampling      Practical Sampling 2: Zero-Order Hold         Summary



          Pulse-Train Sampling
                               So far we have considered sampling by means of an impulse train.
                               In practice, impulses are difficult to generate, and are often approximated
                               by narrow rectangular pulse train
                                                               ∞                                                   ∞
                                             r(t) =                   p1 (t − kTs ) = p1 (t) ∗                              δ(t − kTs )
                                                          k=−∞                                                   k=−∞

                               where p1 (t) represents a rectangular pulse of duration τ << Ts , given by
                                                                                                            t
                                                                               p1 (t) = rect
                                                                                                            τ

                               The sampled signal is then
                                                              xs (t) = x(t)r(t)
                                                                                                       ∞
                                                                       = x(t) p1 (t) ∗                          δ(t − kTs )
                                                                                                    k=−∞
                                                                                      ∞
                                                                       = x(t)                  p1 (t − kTs )
                                                                                  k=−∞

         ENGR201, Lecture 13 — Sampling                                                                                                          Wed, 9 Sept 2009   19/29
1: RPU
          Overview          Ideal Sampling   Reconstruction        Practical Sampling 1: Pulse-Train Sampling      Practical Sampling 2: Zero-Order Hold         Summary



          Pulse-Train Sampling
    QC: RPU/XXX   T1: RPU
25, 2007         19:8



Discrete-time signals and systems
                                                                                           ∞
                      Part III Discrete-time signals and systems
                                                               xs (t) = x(t)                       ∞
                                                                                                    p1 (t − kTs )
                     x(t)                                                             r(t) = ∑
                                                                                     k=−∞ p(t −kTs)
                                                                                                   k = −∞
 ain                                                                                                                                    ∞

ulse-train                                             x(t)                                                                 r(t) = ∑ p(t −kTs)
                                                                                                                                        k = −∞
  nal.                                             t                                                                    t
 (t ); (b) pulse 0                                            −3Ts −2Ts −Ts            0       Ts       2Ts 3Ts
ed signal                                                                              t                                                                     t
                                                         (b)
                                                   0                                            −3Ts −2Ts −Ts               0       Ts       2Ts 3Ts
                                                                                           xs(t)
                       (a)                                                                   (b)

                                                                                                                                xs(t)
                                                                                                                        t
                                                              −3Ts −2Ts −Ts            0       Ts       2Ts 3Ts
                                                                                                                                                             t
                                                         (c)
                                                                                                −3Ts −2Ts −Ts               0       Ts       2Ts 3Ts

         ENGR201, Lecture 13 — Sampling                                                      (c)                                                 Wed, 9 Sept 2009   20/29
     Overview           Ideal Sampling            Reconstruction           Practical Sampling 1: Pulse-Train Sampling            Practical Sampling 2: Zero-Order Hold   Summary



     Pulse-Train Sampling

                             In order to analyze the effect of practical sampling, we need compute the
                             xs (t) = x(t)r(t), which in frequency domain is equivalent to
                                                                                                      1
                                                                                Xs (ω) =                X(ω) ∗ R(ω)
                                                                                                     2π
                             where r(t)                      R(ω)
                             One can show that the FT of r(t) is

                                                                    ∞
                                                                                                                                      ωs τ      nωs τ
                                        R(ω) = 2π                            αn δ(ω − nωs )                    where αn =                  sinc
                                                                 n=−∞
                                                                                                                                      2π         2π

                             Substituting gives
                                                                                  ∞
                                                                                          ωs τ      nωs τ
                                                            Xs (ω) =                           sinc                              X(ω − nωs )
                                                                               n=−∞
                                                                                          2π         2π

P1: RPU/XXX       P2: RPU/XXX              QC: RPU/XXX           T1: RPU
CUUK852-Mandal & Asif  May 25, 2007                 19:8
    ENGR201, Lecture 13 — Sampling                                                                                                                       Wed, 9 Sept 2009   21/29

     Overview           Ideal Sampling            Reconstruction           Practical Sampling 1: Pulse-Train Sampling            Practical Sampling 2: Zero-Order Hold   Summary



     Pulse-Train Sampling
              407                                   9 Sampling and quantization


              Fig. 9.9. Frequency-domain                                                            X(w)
              illustration of the pulse-train
                                                                                                    1
              sampling of a CT signal.
              Spectrum of (a) the original
                                                                                                                                                w
              signal x(t ); (b) the pulse train                                          −2pb 0         2pb
              r(t ); (c) the sampled signal
              xs (t ) = x(t )r(t ).                 (a)
                                                                                                    R(w)


                                                          −3ws       −2ws                                                  2ws        3ws
                                                                                                                                                w
                                                                                   −ws   −2pb   0       2pb    ws

                                                    (b)
                                                                                                    Xs (w) with ws > 2pb


                                                          −3ws       −2ws                                                  2ws        3ws
                                                                                                                                                w
                                                                                   −ws   −2pb   0       2pb    ws

                                                    (c)
                             Thus, in the case of pulse-train sampling, the resulting spectrum Xs (ω)
                             consists of shifted replicas of X(ω) attanuated by a factor αn .
                                        Based on Eq. (9.21b), Fig. 9.9 illustrates the frequency-domain interpretation
                                        of the pulse-train the original signal of the is accomplished
                             The reconstruction of sampling. The spectrum X (ω) x(t) original signal x(t) is by passing
                             xs (t) through anFig. 9.9(a),LPF the spectrum R(ω) of the pulse train r (t)of shown= ωs /2 and a
                                        shown in ideal while with a cut-off frequency is ω
                                                                                                                     c
                                        in Fig. 9.9(b). The spectrum X s (ω) of the sampled signal xs (t) is obtained
                                        by in the pass-band.
                             gain of 1/D0convolving X (ω) with R(ω). As shown in Fig. 9.9(c), X s (ω) consists of
                                                the replicas of X rate fs satisfies the n . Compared to
                             Assuming that shiftedsampling (ω) attenuated with a factor of DNyquist criterion
                                        several
                                        the impulse-train sampling, the spectra of the two sampled signals are identical
                                          and that we use an factor Dn introduced by the pulse-train
                             (fs > 2β),except for a varying attenuation idealofLPF, pulse-train sampling does not
                             introduce any aliasing.
                                        sampling.
     ENGR201, Lecture 13 — Sampling                   Reconstruction of the original signal x(t) from the pulse-train sampled signal                     Wed, 9 Sept 2009   22/29
          Overview         Ideal Sampling         Reconstruction        Practical Sampling 1: Pulse-Train Sampling           Practical Sampling 2: Zero-Order Hold   Summary




                     1     Overview

                     2     Ideal Sampling

                     3     Reconstruction

                     4     Practical Sampling 1: Pulse-Train Sampling

                     5     Practical Sampling 2: Zero-Order Hold

                     6     Summary




          ENGR201, Lecture 13 — Sampling                                                                                                              Wed, 9 Sept 2009   23/29

          Overview         Ideal Sampling         Reconstruction        Practical Sampling 1: Pulse-Train Sampling           Practical Sampling 2: Zero-Order Hold   Summary
X    P2: RPU/XXX            QC: RPU/XXX             T1: RPU
andal & Asif       May 25, 2007            19:8
          Zero-Order Hold Sampling
                               Now, consider another practical sampling method, where we sample a
    408                               Part III Discrete-time signals and systems
                               signal and maintain the value for Ts seconds.
                               This is referred to as sample-and-hold or zero-order-hold
    Fig. 9.10. Time-domain                                             x(t)                                                  xs(t)
    illustration of the zero-order
    hold operation for a CT signal.
    (a) Original signal x(t );
    (b) zero-order hold output x s (t ).                                                        t                                                 t
                                                                   0                                 −3Ts −2Ts −Ts       0      Ts   2Ts 3Ts

                                      (a)                          (b)
                               Note that unlike in pulse-train sampling, the amplitude of the sampled
                               isgnal is maintained constant until the subsequent sample
                                    zero-order hold operation is illustrated to
                               Mathematically, this is equivalent in Fig. 9.10. Unlike the pulse-train sam-
                                          pling, the amplitude of the sampled signal is maintained constant for Ts seconds
                                                     ∞                                                         ∞
                                          until the next sample is taken.
                                                           x(kTs )p2 the zero-order hold (t) ∗ can
                                      xs (t)For mathematical analysis,(t − kTs ) = p2 operationx(t)be modeled by− kTs )
                                              =                                                                       δ(t
                                          the following expression:
                                                  k=−∞                                                      k=−∞
                                                                                          ∞
                                                                              xs (t) =          x(kTs ) p2 (t − kTs )                      (9.22a)
                                                                                         k=−∞
                               where p2 (t) represents a rectangular pulse defined by
                                            or
                                                                    ∞                                    t − 0.5Ts ∞
                                              xs (t) = p2 (t) ∗
                                                                                p2 (t) = rect
                                                                              x(kT )δ(t − kT ) =      p2 (t) ∗ Tx(t)             δ(t − kTs ) ,
                                                                                  s             s
                                                                                                                s
                                                                   k=−∞                                                 k=−∞
                                                                                                                                           (9.22b)
          ENGR201, Lecture 13 — Sampling                                                                                                              Wed, 9 Sept 2009   24/29
Overview     Ideal Sampling                 Reconstruction             Practical Sampling 1: Pulse-Train Sampling                         Practical Sampling 2: Zero-Order Hold   Summary



Zero-Order Hold Sampling
                    Taking the FT results in
                                                                                                                              ∞
                                                                         1
                                                        Xs (ω) = P2 (ω)    X(ω) ∗                                                         δ(ω − kωs )
                                                                        2π
                                                                                                                             k=−∞
                                                                                                 ∞
                                                                    = P2 (ω)                                     X(ω − kωs )
                                                                                          k=−∞

                    where p2 (t)                          P2 (ω)
                    Using a FT table, we have that

                                                                   t − 0.5Ts                                                  ωTs
                                                        rect                                                     Ts sinc                     e−j0.5ωTs
                                                                      Ts                                                      2π
                    Substituting, gives
                                                                                                                                  ∞
                                                                            −j0.5ωTs                               ωTs
                                                   Xs (ω) = e                                sinc                                           X(ω − kωs )
                                                                                                                   2π
                                                                                                                              k=−∞

                    Compare to the equivalent expression for pulse-train sampling - note the ω
                    prior to the summation
ENGR201, Lecture 13 — Sampling                                                                                                                                    Wed, 9 Sept 2009   25/29

Overview     Ideal Sampling                 Reconstruction             Practical Sampling 1: Pulse-Train Sampling                         Practical Sampling 2: Zero-Order Hold   Summary



Zero-Order Hold Sampling
      P1: RPU/XXX       P2: RPU/XXX            QC: RPU/XXX       T1: RPU
      CUUK852-Mandal & Asif           May 25, 2007       19:8

                                                                                                                              ∞
                                                                −j0.5ωTs                                     ωTs
                                      Xs (ω) = e                                      sinc                                                X(ω − kωs )
                    409                                 9 Sampling and quantization                          2π
                                                                                                                             k=−∞
                    Fig. 9.11. Frequency-domain                                                      X(w)
                    illustration of the zero-order
                    hold operation for a CT signal.                                                  1
                    CTFTs of the: (a) original signal
                    x(t ); (b) periodic replicas; and
                                                                                                                                                  w
                    (c) the sampled signal x s (t ).                                      −2pb   0           2pb
                                                         (a)
                                                                                                         ∞

                                                                                                     ∑ X (w − 2kp )
                                                                                                     k = −∞
                                                                                                              Ts




                                                                                                                                                  w
                                                                −2ws           −ws        −2pb   0           2pb        ws          2ws

                                                         (b)

                                                                                                     Xs(w) with ws > 2pb




                                                                                                                                                  w
                                                                −2ws                      −2pb   0           2pb        ws        2ws

                                                         (c)



                                                value of P (ω), Eq. (9.23) can be expressed as follows:
                    Unlike the pulse-trainthesampling, some distortion in amplitude is introduced
                                   Substituting                                       2


                    at the center replica.X (ω) = e           sinc
                                                                     ωT
                                                                        s  ·   −j 0.5 ωTs
                                                                                  X ω−
                                                                                          2kπ
                                                                                                 .
                                                                                                             s
                                                                                                        (9.25)
                                                                                                                    ∞

                                                                                                     2π            k=−∞
                                                                                                                               Ts
ENGR201, Lecture 13 — Sampling                                                                                                                                    Wed, 9 Sept 2009   26/29
Overview       Ideal Sampling   Reconstruction   Practical Sampling 1: Pulse-Train Sampling       Practical Sampling 2: Zero-Order Hold   Summary



Zero-Order Hold Sampling
                  This distortion can be reduced by increasing the width of the main lobe of
                  the sinc function
                  The width of the main lobe is 2π/Ts , and thus increasing its width is
                  equivalent to reducing the sampling interval Ts
                  Due to the distortion, an ideal LPF will recover only an approximate
                  version of x(t)
                  Recalling that we have
                                                                                              ∞
                                                   −j0.5ωTs                  ωTs
                                   Xs (ω) = e                     sinc                              X(ω − kωs )
                                                                             2π
                                                                                        k=−∞

                  we see that perfect reconstruction can be achieved by using a filter given
                  by
                                                      1
                                                sinc(ωTs /2π)
                                                              |ω| ≤ ωs /2
                                     H(ω) =
                                                0             elsewhere
                  This is referred to as the compensation or anti-imaging filter.
                  Note that this filter will still result in an linear phase offset of due to the
                  exponential e−j0.5ωTs - this however corresponds to a delay in the time
                  domain, and is thus not considered distortion.
ENGR201, Lecture 13 — Sampling                                                                                            Wed, 9 Sept 2009   27/29

Overview       Ideal Sampling   Reconstruction   Practical Sampling 1: Pulse-Train Sampling       Practical Sampling 2: Zero-Order Hold   Summary




           1   Overview

           2   Ideal Sampling

           3   Reconstruction

           4   Practical Sampling 1: Pulse-Train Sampling

           5   Practical Sampling 2: Zero-Order Hold

           6   Summary




ENGR201, Lecture 13 — Sampling                                                                                            Wed, 9 Sept 2009   28/29
Overview     Ideal Sampling   Reconstruction   Practical Sampling 1: Pulse-Train Sampling   Practical Sampling 2: Zero-Order Hold   Summary



Summary of Key Ideas
            1   Sampling is a process of discritizing in time a continuous signal (the first
                step in analog-to-digital conversion)
            2   Ideal sampling involves multiplying a function x(t) by an impulse train
                with spacing Ts = 1/fs = 2π/ωs (sampling interval)
                   1  Spectrum of the sampled signal is a series of X(ω) displaced by nωs
                   2  For a signal bandlimited to β Hz, the spectral components of xs (t)
                      do not overlap and perfect reconstruction is possible provided we
                      sample at a rate of fs > 2β (Nyquist rate)
                   3  Otherwise aliasing results
            3   In practice, we cannot generate an ideal impulse, instead we sample by a
                pulse-train (narrow pulses)
                   1  Sampling with an pulse-train results a repeated spectrum but with
                      scaled spectral replicas
                   2  Perfect reconstruction is possible
            4   Sampling using a sample-and-hold circuit is similar to pulse-train
                sampling, but with no ’spacing’ between pulses
                   1  Each spectral replica is distorted, using an ideal reconstruction filter
                      would results in distortion
                   2  Compensation filter must be used to ’reverse’ the spectral distortion
ENGR201, Lecture 13 — Sampling                                                                                      Wed, 9 Sept 2009   29/29

				
DOCUMENT INFO
Shared By:
Categories:
Tags:
Stats:
views:5
posted:11/27/2011
language:English
pages:15