Chap 4 Sampling of Continuous-Time Signals Introduction 4.1 Periodic Sampling 4.2 Frequency-Domain Representation of Sampling 4.3 Reconstruction of a Bandlimited Signal from its Samples 4.4 Discrete-Time Processing of Continuous-Time Signals 4.5 Continuous-Time Processing of Discrete-Time Signals 4.6 Changing the Sampling Rate Using Discrete-Time Processing 4.7 Multirate Signal Processing 4.8 Digital Processing of Analog Signals 4.9 Oversampling and Noise Shaping in A/D and D/A Conversion 2018/9/18 DSP 2 Periodic Sampling Sequence of samples x[n] is obtained from a continuous-time signal xc(t) : x[n] = xc(nT), - infinity < n < infinity T : sampling period fs = 1/T : sampling frequency, samples/second C/D Continuous-to-discrete-time Xc(t) converter X[n] = Xc(nT) T In a practical setting, the operation of sampling is often implemented by an analog-to-digital (A/D) converter which can be approximated to the ideal C/D converter. The sampling operation is generally not invertible. The inherent ambiguity in sampling is of primary concern in signal processing. 2018/9/18 DSP 3 Sampling with a Periodic Impulse Train s(t) C/D Converter Conversion from impulse train x[n] = x (nT) x (t) X C C xS(t)to discrete-time sequence xC(t) xC(t) xS(t) 2 Sampling Rates xS(t) -4T 0 2T -4T 0 2T x[n] 2 Output Sequences x[n] -4 0 2 -4 0 2 2018/9/18 DSP 4 Frequency-domain representation of sampling The conversion of xc(t) to xs(t) through modulating signal s(t) which is a periodic impulse train s(t) (t nT ) n xs (t) xc (t)s(t) xc (t) (t nT ) n by shifting property of the impulse xs (t) xc (nT)(t nT) n The Fourier transform of a periodic impulse train is a periodic impulse train. 2 {s(t)} S( j) ( ks ) T k where s 2/ T : sampling frequency in radians/sec 2018/9/18 DSP 5 Frequency-domain representation of sampling 1 1 X s ( j) X c ( j)S( j) X c ( j kjs ) 2 T k xc(t) can be recovered from xs(t) with an ideal lowpass filter with frequency response Hr(j ). Xr(j ) = Hr(j )Xs(j ) where s > 2 N If Hr(j ) is an ideal lowpass filter with gain T and cutoff frequency such that N < c < ( s - N ) then Xr(j ) = Xc(j ) ALIASING is the distortion in reconstruction process due to s 2 N Nyquist Sampling Theorem : Let xc(t) be a bandlimited signal with Xc(j ) = 0 for | | > N . Then xc(t) is uniquely determined by its samples xc[n] = xc(nT), n = 0, 1,... if > 2 N s 2/ T N: Nyquist frequency and 2 N: Nyquist rate 2018/9/18 DSP 6 Nyquist Sampling Theorem 2018/9/18 DSP 7 Effect in the frequency domain of sampling in the time domain XC(j) 1 Spectrum of the original -N N S(j) Spectrum of the sampling function 2/T -S 0 S X (j) S = 2N Aliasing Effect S > 2 S N S < 2N -S -N N S Spectrum of the sampled signal 2018/9/18 DSP 8 2018/9/18 DSP 9 Example 4.1 If we sample the continuous-time signal xc ( t ) cos(4000 t ) with a sampling period T=1/6000 In this case, s 2/T 12000 The conditions of the Nyquist sampling theorem are satisfied. The Fourier transform of xc(t) is Xjc()()() 4000 4000 2018/9/18 DSP 10 Figure 4.6 (a) Continuous-time and (b) discrete-time Fourier transforms for sampled cosine signal with frequency Ω0 = 4000π and sampling period T = 1/6000. Relation between Continuous and Discrete- Time Domains 2018/9/18 DSP 12 Relation between Continuous and Discrete-Time Domains 2018/9/18 DSP 13 Relation between Continuous and Discrete-Time Domains 2018/9/18 DSP 14 Aliasing 2018/9/18 DSP 15 Frequency-domain representation of sampling 3 The discrete Fourier transform of the sequence x[n] is X(e j ) x[n]e jn n where x[n] = xc(nT). But jTn X s ( j) {xs (t)} { xc (nT) (t nT)} xc (nT)e n n so 1 X ( j) X(e j )| X(e jT ) X ( j jk ) s T T c s or k j 1 2k X(e ) Xc ( j j ) T k T T 2018/9/18 DSP 16 2018/9/18 DSP 17 Reconstruction of a Bandlimited Signal from its samples Samples of a continuous-time bandlimited signal taken frequently enough are sufficient to represent the signal exactly in the sense that the signal can be recovered from the samples and from the knowledge of the sampling period. If Hr(j W) has gain T and cutoff frequency pi/T then sin(t / T) h (t) r t / T and sin[(t nT) / T] xr (t) x[n]hr (t nT) x[n] n n (t nT) / T This means that the ideal lowpass filter interpolates between the impulses of xs(t) to construct a continuous-time signal xr(t). jTn jT Xr ( j) x[n]Hr ( j)e Hr ( j)X(e ) n 2018/9/18 DSP 18 Block diagram, frequency response, and impulse response of an ideal bandlimited signal reconstruction system Ideal reconstruction system Convert the Ideal xS(t) sequence to reconstruction x[n] impulse train filter Hr(j) x r (t) Sampling period T H (j) h r(t) r 1 T t -/T /T -3T -T 0 T 3T 2018/9/18 DSP 19 2018/9/18 DSP 20 Ideal bandlimited interpolation 2018/9/18 DSP 21 Discrete-time processing of continuous-time signal 2018/9/18 DSP 22 Discrete-Time Processing of Continuous- Time Signals 2 C/D converter produces a discrete-time signal x[n] xc (nT) j 1 2k X(e ) Xc ( j j ) T k T T Linear Time-Invariant Discrete-Time Systems Y(ej) = H(ej)X(ej) jT jT and Yr(j) = Hr(j)H(e )X(e ) with = /T D/C converter creates a continuous-time output signal sin[(t nT) / T] yr (t) y[n] n (t nT) / T jT Yr ( j) Hr ( j)Y(e ) 2018/9/18 DSP 23 2018/9/18 DSP 24 Example of Ideal Continuous-Time Lowpass Filtering Using a Discrete-Time Lowpass Filter a) Fourier transform of a bandlimited input signal b) Fourier transform of the sampled input c) Discrete-time Fourier transform of sequence of samples and frequency response of the discrete-time system d) Fourier transform of output of the discrete-time system e) Fourier transform of output of the discrete-time system and frequency response of ideal reconstruction filter f) Fourier transform of output 2018/9/18 DSP 25 Discrete-Time Processing of Continuous-Time Signals Example 4.4 Discrete-Time Implementation of an Ideal Continuous-Time Bandlimited Differentiator d eq. 4.43 y ( t ) x ( t ) ccdt eq. 4.44 Hc ( j ) j Discrete-Time Processing of Continuous-Time Signals jT ,/ eq.4.45 Hjeff ( ) 0, / T j eq. 4.46 He (j ) , T 0,n 0 eq. 4.47 hn cos n ,0n nT Figure 4.13 (a) Frequency response of a continuous-time ideal bandlimited differentiator Hc (jΩ) = jΩ, |Ω| < π/T. (b) Frequency response of a discrete-time filter to implement a continuous-time bandlimited differentiator. Discrete-Time Processing of Continuous- Time Signals: Impulse Invariance The LTI behavior of the system depends on 2 factors: discrete-time system must be linear and time-invariant input signal must be bandlimited and the sampling rate must be high enough so that any aliased components are removed by the discrete-time system Discrete-time system is said to be an impulse- invariant version of the continuous-time system when h[n] = Thc(nT) j and H(e ) = Hc(j/T), || < and = 2018/9/18 DSP 29 Impulse Invariance Figure 4.14 (a) Continuous-time LTI system. (b) Equivalent system for bandlimited inputs. 30 Impulse Invariance j eq. 4.48 H ( e ) Hc ( j / T ), eq. 4.49 Hc ( j ) 0, / T eq. 4.50 h n Thc ( nT ) 31 Impulse Invariance From sampling depicted in Eq. (4.16) eq. 4.51 h n hc ( nT ) j 12k eq. 4.52 H ( e ) Hc j TTTk j 1 eq. 4.53 H ( e ) Hc j , TT Modifying Eqs (4.51) and (4.53) to account for the scale factor of T, we have eq. 4.54 h n Thc ( nT ) j eq. 4.55 H ( e ) Hc j , T 32 Impulse Invariance Example 4.5 A Discrete-time Lowpass Filter obtained by Impulse Invariance 1, c Hjc () 0, c sin( t ) ht() c c t sin( nT ) sin( n ) h n Th() nT T cc c nT n 1, j c He() 0, c 33 Continuous-Time Processing of Discrete- Time Signals 2018/9/18 DSP 34 2018/9/18 DSP 35 Continuous-Time Processing of Discrete- Time Signals For ideal D/C : Xc(jW) and Yc(jW) are zero for | | /T We can express D/C as follows. sin[(t nT) / T] xc (t) x[n] n (t nT) / T sin[(t nT) / T] yc (t) y[n] n (t nT) / T where x[n] = xc(nT) and y[n] = yc(nT) , and frequency-domain : jT Xc ( j) TX(e ),........... / T Yc ( j) Hc ( j)Xc ( j),.... / T 1 Y(e j ) Y ( j / T),.......... T c Overall system behaves as a discrete-time system whose j H(e ) Hc ( j / T),.....