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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 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

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