# ENGR201 Lecture 13 — Sampling

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```					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
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

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