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1、2020/7/17,1,Phase and Group Delays相位和群延迟,The output yn of a frequency-selective LTI discrete-time system with a frequency response exhibits some delay relative to the input xn caused by the nonzero phase response of the system For an input,2020/7/17,2,Phase and Group Delays相位和群延迟,the output is Thus,
2、 the output lags in phase by radians Rewriting the above equation we get,2020/7/17,3,Phase and Group Delays相位和群延迟,This expression indicates a time delay, known as phase delay相位延迟, at given by Now consider the case when the input signal contains many sinusoidal components with different frequencies t
3、hat are not harmonically related,2020/7/17,4,Phase and Group Delays相位和群延迟,In this case, each component of the input will go through different phase delays when processed by a frequency-selective LTI discrete-time system Then, the output signal, in general, will not look like the input signal The sig
4、nal delay now is defined using a different parameter,2020/7/17,5,Phase and Group Delays相位和群延迟,To develop the necessary expression, consider a discrete-time signal xn obtained by a double-sideband suppressed carrier (DSB-SC) modulation(双边带抑制载波（DSB-SC调制)with a carrier frequency of a low-frequency sinu
5、soidal signal of frequency :,2020/7/17,6,Phase and Group Delays相位和群延迟,The input can be rewritten as where and Let the above input be processed by an LTI discrete-time system with a frequency response satisfying the condition,2020/7/17,7,Phase and Group Delays相位和群延迟,The output yn is then given by Not
6、e: The output is also in the form of a modulated carrier signal with the same carrier frequency and the same modulation frequency as the input,2020/7/17,8,Phase and Group Delays相位和群延迟,However, the two components have different phase lags relative to their corresponding components in the input Now co
7、nsider the case when the modulated input is a narrowband signal with the frequencies and very close to the carrier frequency , i.e. is very small,2020/7/17,9,Phase and Group Delays相位和群延迟,In the neighborhood of we can express the unwrapped phase response as by making a Taylors series expansion and ke
8、eping only the first two terms Using the above formula we now evaluate the time delays of the carrier and the modulating components,2020/7/17,10,Phase and Group Delays相位和群延迟,In the case of the carrier signal载波信号we have which is seen to be the same as the phase delay if only the carrier signal is pas
9、sed through the system,2020/7/17,11,Phase and Group Delays相位和群延迟,In the case of the modulating component调制分量we have The parameter is called the( group delay 群延迟)caused by the system at,2020/7/17,12,Phase and Group Delays相位和群延迟,The group delay is a measure of the linearity of the phase function as a
10、function of the frequency It is the time delay between the waveforms of underlying continuous-time signals whose sampled versions, sampled at t = nT, are precisely the input and the output discrete-time signals,2020/7/17,13,Phase and Group Delays相位和群延迟,If the phase function and the angular frequency
11、 w are in radians per second, then the group delay is in seconds Figure below illustrates the evaluation of the phase delay and the group delay,2020/7/17,14,Phase and Group Delays相位和群延迟,Figure below shows the waveform of an amplitude-modulated input and the output generated by an LTI system,2020/7/1
12、7,15,Phase and Group Delays相位和群延迟,Note: The carrier component at the output is delayed by the phase delay and the envelope of the output is delayed by the group delay relative to the waveform of the underlying continuous-time input signal The waveform of the underlying continuous-time output shows d
13、istortion when the group delay of the LTI system is not constant over the bandwidth of the modulated signal,2020/7/17,16,Phase and Group Delays相位和群延迟,If the distortion is unacceptable, a delay equalizer is usually cascaded with the LTI system so that the overall group delay of the cascade is approxi
14、mately linear over the band of interest To keep the magnitude response of the parent LTI system unchanged the equalizer must have a constant magnitude response at all frequencies,2020/7/17,17,Phase and Group Delays相位和群延迟,Example - The phase function of the FIR filter is Hence its group delay is give
15、n by verifying the result obtained earlier by simulation,2020/7/17,18,Phase and Group Delays相位和群延迟,Example - For the M-point moving-average filter the phase function is Hence its group delay is,2020/7/17,19,Frequency Response of the LTI Discrete-Time SystemLTI离散系统的频率响应,The convolution sum descriptio
16、n of the LTI discrete-time system is given by Taking the DTFT of both sides we obtain,2020/7/17,20,Frequency Response of the LTI Discrete-Time System LTI离散系统的频率响应,Or,2020/7/17,21,Frequency Response of the LTI Discrete-Time System LTI离散系统的频率响应,Hence, we can write In the above is the frequency respons
17、e of the LTI system The above equation relates the input and the output of an LTI system in the frequency domain,2020/7/17,22,Frequency Response of the LTI Discrete-Time System LTI离散系统的频率响应,It follows from the previous equation For an LTI system described by a linear constant coefficient difference
18、equation of the form we have,2020/7/17,23,4.3 The Transfer Function传递函数,A generalization of the frequency response function The convolution sum description of an LTI discrete-time system with an impulse response hn is given by,2020/7/17,24,The Transfer Function传递函数,Taking the z-transforms of both si
19、des we get,2020/7/17,25,The Transfer Function传递函数,Or, Therefore, Thus, Y(z) = H(z)X(z),2020/7/17,26,The Transfer Function传递函数,Hence, The function H(z), which is the z-transform of the impulse response hn of the LTI system, is called the transfer function or the system function系统函数The inverse z-trans
20、form of the transfer function H(z) yields the impulse response hn,2020/7/17,27,The Transfer Function传递函数,Consider an LTI discrete-time system characterized by a difference equation Its transfer function is obtained by taking the z-transform of both sides of the above equation Thus,2020/7/17,28,The T
21、ransfer Function传递函数,Or, equivalently as An alternate form of the transfer function is given by,2020/7/17,29,The Transfer Function传递函数,Or, equivalently as are the finite zeros零点, and are the finite poles极点 of H(z) If N M, there are additional zeros at z = 0 If N M, there are additional poles at z =
22、0,2020/7/17,30,The Transfer Function传递函数,For a causal IIR digital filter, the impulse response is a causal sequence The ROC of the causal transfer function is thus exterior to a circle going through the pole farthest from the origin Thus the ROC is given by,2020/7/17,31,The Transfer Function传递函数,Exa
23、mple - Consider the M-point moving-average FIR filter with an impulse response Its transfer function is then given by,2020/7/17,32,The Transfer Function传递函数,The transfer function has M zeros on the unit circle at , There are poles at z = 0 and a single pole at z = 1 The pole at z = 1 exactly cancels
24、 the zero at z = 1 The ROC is the entire z-plane except z = 0,M = 8,2020/7/17,33,Exercise 5.1.3. Producing Pole/Zero Plots.,clc; clear; close all; % Step (a). b = 1 3 3 1; % Numerator Coefficients. a = 1 .5 .3 .1; % Denominator Coefficients. % Produce and display the Poles/Zeros plot. figure(Name,Ex
25、ercise 5.1.3. Producing Pole/Zero Plots); zplane(b,a); grid on; % Produce and display the frequency response. figure(Name,Exercise 5.1.3. Producing Pole/Zero Plots); freqz(b,a); % or better: % fvtool(b,a); % Step (b). % Find the numerical values of poles and zeros z p k = tf2zpk(b,a),2020/7/17,34,re
26、sult,2020/7/17,35,The Transfer Function传递函数,Example - A causal LTI IIR digital filter is described by a constant coefficient difference equation given by Its transfer function is therefore given by,2020/7/17,36,The Transfer Function传递函数,Alternate forms: Note: Poles farthest from z = 0 have a magnitu
27、de ROC:,2020/7/17,37,Frequency Response from Transfer Function传递函数的频率响应,If the ROC of the transfer function H(z) includes the unit circle, then the frequency response of the LTI digital filter can be obtained simply as follows: For a real coefficient transfer function H(z) it can be shown that,2020/
28、7/17,38,Frequency Response from Transfer Function传递函数的频率响应,For a stable rational transfer function in the form the factored form of the frequency response is given by,2020/7/17,39,Frequency Response from Transfer Function传递函数的频率响应,It is convenient to visualize the contributions of the zero factor an
29、d the pole factor from the factored form of the frequency response The magnitude function is given by,2020/7/17,40,Frequency Response from Transfer Function传递函数的频率响应,which reduces to The phase response for a rational transfer function is of the form,2020/7/17,41,% Exercise Sketching the Magnitude Re
30、sponse.,% Pole ,2020/7/17,42,th = -pi; pt = exp( j*th); xx = real(pt); yy = imag(pt); l1 = line(Xdata,xx,Ydata,yy); set(l1,Color,w,Marker,o,EraseMode,xor); l4 = line(Xdata, real ( pt zz(1) ),Ydata,imag( pt zz(1),Color,b,LineStyle,-,EraseMode,xor); l6 = line(Xdata, real ( 0 pt ),Ydata,imag(0 pt ),Col
31、or,r,LineStyle,-,EraseMode,xor); M = 128; w = -pi:pi/M:pi; hh = freqz(b,a,w); h = abs(hh); ang = unwrap(angle(hh)/pi*180; angmax = max(ang); angmin = min(ang); maxh = max(h); if min(h) 0 minh = min(h); else minh = 0; end,2020/7/17,43,subplot(2,2,2) title(|H(jomega)|); xx = w(1) w(1) ; yy = h(1) h(1)
32、 ; ll1 = line (Xdata,xx,Ydata,yy,Color,y,LineStyle,-,Erasemode,none); ll2 = line (Xdata,w(1),Ydata,h(1),Color,w,Marker,o,Erasemode,xor); set(gca,XTick,-pi:pi/4:pi); set(gca,XTickLabel,-pi,-3pi/4,-pi/2,-pi/4,0,pi/4,pi/2,3pi/4,pi ); axis ( -pi pi minh maxh ); grid ylabel (gain) xlabel (frequency omega
33、 (rad/sample); hold on subplot(2,2,4) title(angleH(jomega); xx = w(1) w(1) ; yy = ang(1) ang(1) ; lx1 = line (Xdata,xx,Ydata,yy,Color,y,LineStyle,-,Erasemode,none); lx2 = line (Xdata,w(1),Ydata,h(1),Color,w,Marker,o,Erasemode,xor); set(gca,XTick,-pi:pi/4:pi); set(gca,XTickLabel,-pi,-3pi/4,-pi/2,-pi/
34、4,0,pi/4,pi/2,3pi/4,pi ); axis ( -pi pi angmin angmax ); grid ylabel (phase (degrees) xlabel (frequency omega (rad/sample); hold on step = 2*pi/M;,2020/7/17,44,for ii = 1:4:2*M+2; th = -pi + (ii-1)*step/2; pt = exp( j*th); set(l1,Xdata,real(pt),Ydata,imag(pt); set(l4,Xdata, real ( pt zz(1) ),Ydata,i
35、mag( pt zz(1); set(l6,Xdata, real (0 pt ),Ydata,imag(0 pt ); if ii=1, xx = w(1) w(1) ; yy = h(1) h(1) ; else xx = w(ii-1:ii); yy = h(ii-1:ii); end set(ll1,Xdata,w(1:ii),Ydata,h(1:ii); set(ll2,Xdata,w(ii),Ydata,h(ii); if ii=1, xx = w(1) w(1) ; yy = ang(1) ang(1) ; else xx = w(ii-1:ii); yy = ang(ii-1:
36、ii); end set(lx1,Xdata,w(1:ii),Ydata,ang(1:ii); set(lx2,Xdata,w(ii),Ydata,ang(ii); pause(0.25) end,2020/7/17,45,result,2020/7/17,46,% Exercise 5.4.1.b. Sketching the Magnitude Response.,% Exercise 5.4.1.b. Sketching the Magnitude Response. % Pole hold on,2020/7/17,47,fprintf(1,Exercise 5.4.1.b.: pol
37、e/zero plot - press return to animaten) pause th = -pi; pt = exp( j*th); xx = real(pt); yy = imag(pt); l1 = line(Xdata,xx,Ydata,yy); set(l1,Color,w,Marker,o,EraseMode,xor); l2 = line(Xdata, real ( pt pp(1) ),Ydata,imag( pt pp(1),Color,g,LineStyle,-,EraseMode,xor); % l3 = line(Xdata, real ( pt pp(2)
38、),Ydata,imag( pt pp(2),Color,g,LineStyle,-,EraseMode,xor); l4 = line(Xdata, real ( pt zz(1) ),Ydata,imag( pt zz(1),Color,b,LineStyle,-,EraseMode,xor); l6 = line(Xdata, real ( 0 pt ),Ydata,imag(0 pt ),Color,r,LineStyle,-,EraseMode,xor); M = 128; w = -pi:pi/M:pi; hh = freqz(b,a,w); h = abs(hh); ang =
39、unwrap(angle(hh)/pi*180; angmax = max(ang); angmin = min(ang); maxh = max(h); if min(h) 0 minh = min(h); else minh = 0; end,2020/7/17,48,subplot(2,2,2) title(|H(jomega)|); xx = w(1) w(1) ; yy = h(1) h(1) ; ll1 = line (Xdata,xx,Ydata,yy,Color,y,LineStyle,-,Erasemode,none); ll2 = line (Xdata,w(1),Ydat
40、a,h(1),Color,w,Marker,o,Erasemode,xor); set(gca,XTick,-pi:pi/4:pi); set(gca,XTickLabel,-pi,-3pi/4,-pi/2,-pi/4,0,pi/4,pi/2,3pi/4,pi ); axis ( -pi pi minh maxh ); grid ylabel (gain) xlabel (frequency omega (rad/sample); hold on subplot(2,2,4) title(angleH(jomega); xx = w(1) w(1) ; yy = ang(1) ang(1) ;
41、 lx1 = line (Xdata,xx,Ydata,yy,Color,y,LineStyle,-,Erasemode,none); lx2 = line (Xdata,w(1),Ydata,h(1),Color,w,Marker,o,Erasemode,xor); set(gca,XTick,-pi:pi/4:pi); set(gca,XTickLabel,-pi,-3pi/4,-pi/2,-pi/4,0,pi/4,pi/2,3pi/4,pi ); axis ( -pi pi angmin angmax ); grid ylabel (phase (degrees) xlabel (fre
42、quency omega (rad/sample); hold on,2020/7/17,49,step = 2*pi/M; for ii = 1:4:2*M+2; th = -pi + (ii-1)*step/2; pt = exp( j*th); set(l1,Xdata,real(pt),Ydata,imag(pt); set(l2,Xdata, real ( pt pp(1) ),Ydata,imag( pt pp(1); % set(l3,Xdata, real ( pt pp(2) ),Ydata,imag( pt pp(2); set(l4,Xdata, real ( pt zz
43、(1) ),Ydata,imag( pt zz(1); set(l6,Xdata, real (0 pt ),Ydata,imag(0 pt ); if ii=1, xx = w(1) w(1) ; yy = h(1) h(1) ; else xx = w(ii-1:ii); yy = h(ii-1:ii); end set(ll1,Xdata,w(1:ii),Ydata,h(1:ii); set(ll2,Xdata,w(ii),Ydata,h(ii); if ii=1, xx = w(1) w(1) ; yy = ang(1) ang(1) ; else xx = w(ii-1:ii); y
44、y = ang(ii-1:ii); end set(lx1,Xdata,w(1:ii),Ydata,ang(1:ii); set(lx2,Xdata,w(ii),Ydata,ang(ii); pause(0.25) end,2020/7/17,50,result,2020/7/17,51,Exercise 5.4.2.c. Sketching Magnitude Responses from Pole/Zero Plots.,% Exercise 5.4.2.c. Sketching Magnitude Responses from Pole/Zero Plots. % Pole hold o
45、n fprintf(1,Exercise 5.4.2.c.: pole/zero plot of H3(z) - press return to continuen) pause,2020/7/17,52,th = -pi; pt = exp( j*th); xx = real(pt); yy = imag(pt); l1 = line(Xdata,xx,Ydata,yy); set(l1,Color,w,Marker,o,EraseMode,xor); l2 = line(Xdata, real ( pt pp(1) ),Ydata,imag( pt pp(1),Color,g,LineSt
46、yle,-,EraseMode,xor); l4 = line(Xdata, real ( pt zz(1) ),Ydata,imag( pt zz(1),Color,b,LineStyle,-,EraseMode,xor); l6 = line(Xdata, real ( 0 pt ),Ydata,imag(0 pt ),Color,r,LineStyle,-,EraseMode,xor); M = 128; w = -pi:pi/M:pi; hh = freqz(b,a,w); h = abs(hh); ang = unwrap(angle(hh)/pi*180; angmax = max
47、(ang); angmin = min(ang); if max(h)= Inf maxh = max(h); else ind = find(h=Inf); maxh = max(h(1:ind-1) h(ind+1:end); end,2020/7/17,53,if min(h) 0 minh = min(h); else minh = 0; end subplot(2,2,2); title(|H_3(jomega)|); xx = w(1) w(1) ; yy = h(1) h(1) ; ll1 = line (Xdata,xx,Ydata,yy,Color,y,LineStyle,-
48、,Erasemode,none); ll2 = line (Xdata,w(1),Ydata,h(1),Color,w,Marker,o,Erasemode,xor); set(gca,XTick,-pi:pi/4:pi); set(gca,XTickLabel,-pi,-3pi/4,-pi/2,-pi/4,0,pi/4,pi/2,3pi/4,pi ); axis ( -pi pi minh maxh ); grid ylabel (gain) xlabel (frequency omega (rad/sample); hold on,2020/7/17,54,subplot(2,2,4) t
49、itle(angleH_3(jomega); xx = w(1) w(1) ; yy = ang(1) ang(1) ; lx1 = line (Xdata,xx,Ydata,yy,Color,y,LineStyle,-,Erasemode,none); lx2 = line (Xdata,w(1),Ydata,h(1),Color,w,Marker,o,Erasemode,xor); set(gca,XTick,-pi:pi/4:pi); set(gca,XTickLabel,-pi,-3pi/4,-pi/2,-pi/4,0,pi/4,pi/2,3pi/4,pi ); axis ( -pi pi angmin angmax ); grid ylabel (phase (degrees) xlabel (frequency omega (rad/sample)