Answers to Problems of Chapter 4

Transcription

1 Answers to Problems of Chapter 4 The answers to the problems of this chapter are based on the use of MATLAB. Thus, if the readers have some prior elementary knowledge on it, it will be easier for them to follow the analysis made. We would suggest to the readers that have never worked with MATLAB, first to practice on it in order to become familiar with the most common commands. 4. Using MATLAB find the poles and zeros of an Inverse Chebyshev high-pass NTF, like one of those given in Table 4.4 (specific order and parameter M). Determine the in-band attenuation so that the desired M is achieved. Draw the plot of M as a function of the inband attenuation. Is it verified, this way, that decreasing P e in results in increasing P e tot? Answer: The relevant command of MATLAB is cheby2. The help facility of MATLAB provides online information for most topics. Entering the command help cheby2 in the MATLAB command window you can find more details about the cheby2 command. Then, you can find out that by writing: [z,p,k] = cheby2(n,r,wn,'high') the zeros z and the poles p of an Inverse Chebyshev Transfer Function can be obtained. The argument N determines the order of the Transfer Function, R is the stopband ripple (i.e. R decibels down), W is the cutoff frequency normalized to unity ( corresponds to half the n sample rate, i.e. f s 2 ) while the string high determines that the designed Transfer Function will be high-pass. Note that for low-pass Transfer functions, W is the frequency that the stopband starts, while for high-pass Transfer functions W is the frequency that the stopband ends. The scalar value k, which is also returned, is in order for the maximum gain in the passband to be unity. Consequently, entering the command [z,p,k] = cheby2(5,,/64,'high') the zeros and the poles of a 5 th -order high-pass Inverse Chebyshev Transfer Function with stopband ripple - can be determined. These are. n n

2 z = i i i i p = i i i i k =.5659 If you want to draw the magnitude response, and also to zoom in the stopband, enter the following commands (use also the help command to find details about each command): [b,a] = cheby2(5,,/64,'high'); % Coefficients of Numerator and Denominator [h,w]=freqz(b,a,496); % Determine the Frequency Response subplot(2,,) plot(w/pi, 2*log(abs(h)), 'w') % Plot magnitude response (entire band) axis([ -4 ]) subplot(2,,2) plot(w(:496/8)/pi, 2*log(abs(h(:496/8))), 'w') %Plot magnitude response (pass-band) axis([ /8-4 ]) / Frequency ( normalized f s 2 ) Figure p4.-. Magnitude response of the designed Inverse Chebyshev Transfer function. 2

3 Note that the designed Transfer function is of the form Hz () N ( z z k i = ) (p4.-) ( pi z ) i= Consequently the first term of the impulse response according to eq. (4.6) equals k. Since k is not unity, the designed Transfer function cannot be a Noise Transfer Function. However, The latter can be obtained if we divide the numerator coefficients of the designed Transfer function by k. Thus the parameter M (the maximum value of the magnitude response in the passband, i.e. at z=) will be: M = 2log k = 2log. 5659= (p4.-2) Given the order N and the cutoff frequency W of an Inverse Chebyshev Highpass Transfer Function, the gain k is a function of the stopband ripple R. Since the parameter M is related with k as in eq. (p4.-2), the former can also be determined as a function of the stopband ripple R. As a result, the relationship between M and the in-band attenuation R-M, can also be determined. For a 5 th -order Inverse Chebyshev High-pass Transfer Function this relationship is given by the following diagram: n M () R-M () Figure p4.-2. M as a function of R-M. For example, M=3.5, corresponds to in-band attenuation equal to

4 Let us now examine the relationship between P e in and P e tot. We have Pe in π OSR = Pe NTF e 2π π OSR jθ ( ) 2 dθ (p4.-3) Pe tot = Pe π NTF e 2π π jθ ( ) 2 dθ (p4.-4) π OSR 2 dθ is proportional to π OSR jθ However, NTF( e ) proportional to M ( M R) π 2 dθ is π jθ, while NTF( e ). Moreover from the above plot it is obvious that M is an increasing function of the in-band attenuation R-M. Consequently, as P e in decreases P e tot increases. However this increase depends on the order of the NTF. In the following plot it is shown the dependence of M on R-M for various orders rd -order 4 th -order 5 th -order M () th 7 th 8 th th R-M () Figure p4.-3. M as a function of R-M for various orders. As a result, for a given value R-M of in-band attenuation, if we want the out-band gain not to be high, the order of the Transfer Function will have to be increased. 4

5 4.2 Consider as NTFinitial ( z) one of the Inverse Chebyshev NTFs given in Table 4.4. Write a program in MATLAB for determining, using eq. (4.25), the corresponding NTFs, for various values of k. Find the values of k for which minimization of S h, A and M is achieved. Determine also, the minimum values for S h A and M. Does NTFinitial () z obey some of the criteria given in eqs. (4.22), (4.23) and (4.24)? Answer: In this problem, you will create a Function file which will get as input the zeros and poles of some NTF and will draw the quantities S h, A and M as functions of k and determine their minimum values. Create first a directory, let named Mydir, in the hard-disk of your workstation (for example the C:). In this directory you will save all the files that you will create. Then select from the File menu of the MATLAB window New M-file and write the following: function [Shmin, ksh, Amin, ka, Mmin, km]=ntf2k (z,p) % z, p are the zeros and poles of the initial Noise Transfer Function [NTFnum,NTFden]=zp2tf(z,p,); % Determine the coefficients of numerator and % denominator kl=.7; kh=.6; step=.5; k=[kl:step:kh]; % Set a minimum value for k % Set a maximum value for k for i=:length(k) NTFdenk=k(i)*NTFden+(-k(i))*NTFnum; [H,w]=freqz(NTFnum,NTFdenk,496); M(i)=max(2*log(abs(H))); [h,t]=impz(ntfnum,ntfdenk,5); Sh(i)=sum(abs(h(:5))); A(i)=mean(abs(H).^2); end % Determine the new denominator % Find its frequency response % Find the corresponding M % Find the impulse response coefficients % Find the corresponding Sh % Find the corresponding A 5

6 [Shmin,i]=min(Sh); ksh=k(i); [Amin,i2]=min(A); ka=k(i2); [Mmin,i3]=min(M); km=k(i3); figure() plot(k,sh,'w') figure(2) plot(k,a,'w') figure(3) plot(k,m,'w') % Find the minimum value of Sh % Find the value of k that corresponds to the minimum of Sh % Find the minimum value of A % Find the value of k that corresponds to the minimum of A % Find the minimum value of M % Find the value of k that corresponds to the minimum of M % Plot Sh vs. k % Plot A vs. k % Plot M vs. k end Save this file with the name ntf2k in the directory Mydir. Then enter the following command path(path, c:\mydir ) This is in order to be able to invoke this Function files from the MATLAB command window. are Let choose the 4 th -order E Inverse Chebyshev Transfer Function of Table 4.4. Its poles window {.7839±j.3333,.657±j.57}. For simplicity enter in the MATLAB command [z,p,k] = cheby2(4,94,/64,'high') in order to produce these poles as well as the zeros. Then enter This results in [Shmin, ksh, Amin, ka, Mmin, km]=ntf2k (z,p) Shmin = ksh =.36 Amin = ka =.86 Mmin = 4.99 km = while the following plots are drawn: 6

7 Sh ( k ) S h = k a k=.36 A(k) A= k b k= M(k) M=4.9 k= c Figure p4.2-. Diagrams of S h, A and M as a function of the gain k. k We can observe that the minimum values of the quantities S h, A and M satisfy the criteria given by eqs. (4.22), (4.23) and (4.24). However these minimum values are very near to the bounds so it is expected for the designed modulator to be prone to instability. Indeed the maximum amplitude for stability x max is only.36 as it can be seen in Table

8 4.3 Using SIMULINK design a Σ modulator. Make use of Fig. 4. letting δ = and in place of Lo( z) and L( z) the structure in Fig Determine the values of G(z) and L(z) so that the STF is an all-pass function with the NTF being one of those in Table 4.4. a) For various values of the amplitude of the input signal (always<) and various frequencies, draw the output pulse sequence as well as the input to the quantizer. b) Determine the spectrum of the output using a window of samples and comment on its noise shaping characteristics. c) Evaluate the SNR for various amplitudes of the input signal and draw the corresponding diagram. d) Using the SNR plot, determine the input signal amplitude for which instability occurs. For this amplitude, simulate the modulator and monitor the input and the output of the quantizer. e) Reduce the accuracy of filter coefficients and examine the change in Noise Shaping. f) Use two separate filters Lo( z) and L( z) for the structure in Fig. 4. (instead of G(z) and L(z)) and simulate the resulting modulator. Monitor the signals at their output. Could this structure be suitable for implementation? Answer: Consider the following Σ modulator shown in Fig Its block diagram in signal-flow level is the en ( ) x( n) G() z - L() z ±δ =± y( n) z Figure p4.3-. Block diagram of a Σ modulator. The signal transfer function (STF) will be STF(z)= G(z)L () z + z (p4.3-) L() z while the noise transfer function (NTF) 8

9 For a given NTF the loop filter Lz () NTF(z)= + z L() z can be determined as Lz= () z ( NTF () z ) NTF(z) (p4.3-2) (p4.3-3) while in order for the STF to be all-pass, i.e. STF()=, z it should be Gz= () +z - L(z) (p4.3-4) Consequently, the block diagram becomes: x( n) z en ( ) Lz () - L() z ± y( n) z This can be simplified to Figure p Block diagram of a Σ modulator with unity STF. Lo( z) =+ z L() z x( n) en ( ) z L () z y( n) - L () z = z L() z ± Figure p The Σ modulator block diagram of figure p4.3-2 after the simplification. The filters Lo( z) and L( z) are also shown. However in this structure they are two separate filters. Let us see now how this block diagram can be implemented using the SIMULINK toolbox of MATLAB. 9

10 Step : Invoking SIMULINK Enter the command simulink in the MATLAB command window and the simulink window appears as in the following figure. Figure p Opening MATLAB and invoking SIMULINK. As it can be seen, in the simulink window there are various icons. These are libraries which contain various components. By double clicking on each of them you can see the available components. Step 2: Implementation of the block diagram of the Σ modulator. Make the simulink window active and select with mouse File New. A new window, named untitled appears. This is the model window where you will create the block diagram of the Σ modulator. Start opening the various libraries which exist in the simulink window, and drag and drop the components you need in the model window. You will need the following components: from library Sources a Sine Wave and two Constant, from library Linear two Sum,

11 from library Sinks two To Workspace and a Scope, from library Discrete a Filter and a Unit Delay and from library Nonlinear a switch Then connect them as in the following figure Figure p Implementation of a Σ modulator using SIMULINK. After completing the design, select the save command from the File menu of the model window and give some name, let us say testntf, having also chosen the directory Mydir. The workspace components are needed in order to return the data in the MATLAB workspace for processing. The scope is in order to monitor the input of the quantizer, if we want to do so as in the case of an actual circuit with an oscilloscope. The two Constant components in combination with the Switch component implement the single-bit quantizer. Of course this is not the only way. By combining the various components which are available in the Nonlinear library of SIMULINK it is possible to implement the single-bit quantizer in many different ways.

12 By double clicking in the Filter icon, this opens making possible to insert the coefficients of the numerator and the denominator of Lz (). We can choose any of the NTFs of Tables 4.4 and 4.5. Let choose the 5 th -order E2. For simplicity, return to the MATLAB command window and give the commands [z,p,k] = cheby2(5,,/64,'high') [b,a]=zp2tf(z,p,) The latter is in order to obtain the coefficients of the numerator and denominator of the designed NTF, i.e. NTF(z)= b ( ) + b ( 2) z b ( n + ) z a( ) + a( 2) z a( n+ ) z with normalized such as b() = a () =. Hence, the resulting arrays b and a are n n (p4.3-5) b= a= The coefficients of numerator of Lz () will be the array b, i.e. Lz () can be determined by eq. (p4.3-3). Thus, the coefficients of the will be given by the array b-a, while the coefficients of the denominator Coefficients of numerator of Lz () Coefficients of denominator of Lz ()

13 Step 3: Simulation of the Σ modulator. Simulation of SIMULINK models involves the numerical integration of sets of ordinary differential equations. SIMULINK provides a number of methods for the simulation of such systems which usually are not of fixed time-step. However purely discrete systems can be simulated using any of these methods without any difference in the solutions. Select the Parameters command from the Simulation menu in the simulink window. The Simulink control panel appears as in the following figure. In order to perform a discrete time simulation the parameter Min Step Size should be equal to Max Step Size. Moreover these are set to, while the Stop Time parameter is set to in order to perform a simulation of time-steps. Figure p Setting the parameters in order to perform a simulation in SIMULINK. Set the other parameters as it is shown in the above figure and close this window. Return in the testntf window and by double clicking on the Filter and Unit Delay components, set the Sample Time parameter equal to. This is in order to obtain only data point per time-step of simulation. Then, by double click on the Workspace components in 3

14 the testntf model set the Maximum number of Timesteps parameter equal to This will be the number of data points which will be acquired and returned to MATLAB workspace for processing. The number is convenient since it is a power of 2. Note also that the acquired data points will be the last of the which will be produced during simulation. a) By double clicking on the Sine Wave icon you can set the Amplitude and Frequency parameters equal to the desired values in order to perform simulations for various values of the amplitude and the frequency of the sinusoidal source. You can monitor the various signals by means of the Scope component during simulation. Also, after the simulation has finished, it is possible to draw plots using the command plot of MATLAB. For example, giving the command and plot(u(:6)) plot(y(:6)) you can see the signals u and y during the time interval from to 6 time-steps. The following figure corresponds to sinusoidal input signal with amplitude.45 and frequency 2π rad/time-step. u(n) y(n) Number of Time-steps Figure p Plotting the input and the output of the quantizer. 4

15 b) Let us examine now the spectrum of the output sequence y( n). Enter the following commands: w=blackman(65536); % Blackman Window with points wy=w.*y; cnorm= mean(y.*y)/ mean(wy.^2); % Coefficient for normalization of Signal Power spec=cnorm*(abs(fft(wy/65536)).^2); f=[:65536/2]/65536; figure(); plot(f,*log(spec(:65536/2)),'w') % Plot total Spectrum axis([.5-8 6]); figure(2); plot(f(:65536/2/32),*log(spec(:65536/2/32)),'w') % Plot only inband Spectrum axis([.5/32-8 6]); PSD PSD Inband Frequency a Frequency Figure p The spectrum at the output of the Σ modulator a) entire band, b) inband. Note that the window is needed in order to reduce the edge effects and observe the in-band noise shaping clearly. Since the attenuation of quantization noise is very high in this range of frequencies, any leakage of the signal power to adjacent frequencies can destroy this view. Moreover it is preferable the number of periods of the input signal to be an integer. In the presented example the frequency of the sinusoidal is /4 of the signal band. This equals /(4 2 64) of the sampling rate, or equivalently 28/ Thus, the time window of time-steps contains exactly 28 periods of the input signal. b 5

16 c) In order to draw the SNR plot select from the File menu of the MATLAB window New M-file and write the following commands: amp=[-4:2:-2,-9.5:.5:-8,-7.9:.:-.]; %Amplitude in decibels w=blackman(65536); for k=:length(amp) %------Set Parameters and Perform Simulation vamp=^(amp(k)/2); set_param('testntf/sine Wave','amplitude',vamp); [t,x,y]=linsim('testntf',[,65536],[],[.,,]); %-----Find SNR wy=w.*y; spec=abs(fft(wy)).^2; specin=spec(:65536/2/64); % Inband spectrum i=29; % This index corresponds to the frequency of input signal np=6; % This number is needed in the case that leakage of the signal % power to adjacent frequencies has occurred psignal=sum(specin(i-np:i+np)); % Signal power pnoisein= sum(specin(:i-np-))+ sum(specin(i+np+:65536/2/64)); % Inband noise % power snr(k)=*log(psignal/pnoisein); end if snr(k)< snr(k)=; end plot(amp,snr,,'w') % Plot the SNR vs. amplitude Save this Script file with the name findsnr in the directory Mydir. Then return in the MATLAB command window and enter the command: findsnr Note that the Script file is convenient since you can execute a set of commands at any time without needing to write them again. The following plot is drawn: 6

17 2 SNR Amplitude () Figure p SNR as a function of the input amplitude. Moreover enter the following commands in order to save the data in the directory Mydir. cd c:\mydir save data amp snr d) By means of the above diagram (using the command zoom on in the MATLAB window) you can verify that for input signal amplitude equal to -6.3, or equivalently for x o =. 484, there is a drop in the SNR diagram, while for amplitudes greater than -6 the SNR is zero. However for amplitudes near this range of values, i.e. around -6, the occurrence of instability is very sensitive to the decimal digits of the input signal amplitude, as well as on the frequency of the input signal (perform simulations using other frequencies and examine the results by means of the Scope component). Thus, in order to secure that instability will occur and to observe clearly the transition between stable and unstable operation it is better to choose a slightly greater amplitude, let say x o =52.. Using the plot command you can draw the signal un ( ) in order to see its changes. This is shown in the following figure. Note that almost until the instant 552 the modulator is stable. However, after this time the modulator becomes unstable, i.e. the input of the quantizer starts oscillating with increasing amplitude. Of course in an actual circuit the amplitude of this signal cannot be increased indefinitely as it is shown when simulation is performed. However, the modulator will have stopped modulating the input signal, i.e. the SNR will still be very low. 7

18 u(n) u(n) Number of Time-steps Figure p4.3-. Plots of the quantizer input when the modulator becomes unstable. e) Let us now examine the effect of the accuracy of the filter coefficients. Reduce first the accuracy of the fractional part of the numerator coefficients of Lz () to 2 digits and simulate the modulator. Then draw the spectrum of the output repeating the commands of step b). You must obtain the following plots: y( n) output PSD PSD Inband Frequency a Frequency Figure p4.3-. The spectrum at the output of the Σ modulator when the accuracy of the numerator coefficients of the loop filter is reduced a) entire band, b) inband. b 8

19 We can observe that mainly the in-band spectrum has been affected. However the accuracy of the denominator coefficients is much more significant. So, leave the numerator coefficients unchanged but reduce the accuracy of the fractional part of the denominator coefficients of Lz () to 5 digits. In this case the output spectrum will be PSD PSD Inband a Frequency Frequency Figure p The spectrum at the output of the Σ modulator when the accuracy of the denominator coefficients of the loop filter is reduced a) entire band, b) inband. Of course, these changes in the spectrum are expected since it is known that the position of zeros and poles of high-order filters is very sensitive to the accuracy of the coefficients, and therefore they are usually implemented as a cascade connection of st and 2 nd -order filters. Note also that the loop-filter of a Σ modulator is always implemented as a chain of integrators with distributed feedbacks and local resonator feedbacks as it was explained in Chapter 3, since this structure is more insensitive to the accuracy of the coefficients. b f) Modify the testntf model as in figure p4.3-3 in order for the structure of the modulator to contain two separate filters Lo( z) and L( z), and save this model with another name, say testntf. Then simulate it. Note that no difference to the output appears. However this structure cannot be implemented as an actual circuit. If two separate filters are used, these should be perfectly matched which is not physically possible. Change slightly the coefficients of the denominator of L z ( ) L z ( ) filter and examine the results. Moreover, even in the case that the filters were perfectly matched, the signal amplitude at their outputs would be very high (see the figure p4.3-4 where the signal Lo( z) Lo( z) u( n) is shown in comparison with un ( ) ). This is or or 9

20 because the poles of each of these filters are on the unit circle (the poles of of the NTF). Lz () are the zeros x( n) z L () z Lo () z u n () en ( ) y( n) L z ( ) z L () z - un ( ) u2( n) ± Figure p Block diagram of a modulator with two separate filters Lo( z) and L( z). 4 6 u (n) u(n) Number of Time-steps Figure p Plots of the Lo( z) output and the quantizer input. Furthermore, it is possible due to some imperfection of the actual circuit, the poles of the filter Lo( z) to be outside the unit circle. This would result in unstable operation since u( n) would not be a bounded signal. On the other hand when a single filter is used none of these problems appears. 2

21 4.4 Simulate a Σ modulator whose NTF is one of those in Table 4.4. Find the output spectrum. Can the observed noise shaping characteristic be approximated by an inverse Chebyshev high-pass function? Justify your answer. Answer: We will examine the Σ modulator, which was examined in the previous problem. Let s also perform the simulation with the signal amplitude to be equal to.45 and the frequency to 28/65536, and draw the spectrum of the output as in the following figures: -5-2 PSD Inverse Chebyshev -4 Transformation using ke Frequency - - Signal frequency a PSD Transformation using ke Inverse Chebyshev Frequency b Figure p4.4-. Approximations of the Σ modulator output spectrum a) entire band, b) inband. 2

22 Our purpose is to verify if the quantization noise shaping can be approximated by the Inverse Chebyshev Transfer function which has been chosen in order to design this modulator. If the linear model was valid, according to eqs. (4.7) and (4.8) (with δ =) it should be Pe tot Consequently the quantization noise power should be π Pe jθ 2 = NTF( e ) dθ = Px (p4.4-) 2π π Pe = π π Px NTF e jθ ( ) 2 dθ (p4.4-2) However as it is shown in the above figure the actual noise shaping cannot be approximated well by the high-pass Inverse Chebyshev Transfer Function. Thus, the linear model does not seem accurate. However, it could give better estimation of the quantization noise shaping if we assumed that the gain k before the quantizer has a value different from unity as in the quasi-linear model. According to the latter the output of the modulator is given by eq. (4.), i.e. with the gains k s and k e yn ( ) = ku( n) + ku( n) + en ( ) (p4.4-3) s s e e to be estimated theoretically. However, these gains can also be measured experimentally easily. According to eqs (4.2a) and (4.2b) they are given by k k s e = = { ( ) s( n) } { ( ) ( )} E y n u Eu nu n s s { ( ) e( n) } { ( ) ( )} E y n u E u n u n Since u n s( ) is the signal component, it will be given by while the noise component ue ( n) e e { } { ( ) ( )} (p4.4-4) (p4.4-5) E u n x n us ( n ) ( ) ( ) = xn ( ) (p4.4-6) E x n x n will be { ( ) ( )} { ( ) ( )} E u n x n ue( n) = un ( ) us( n) = un ( ) xn ( ) (p4.4-7) E x n x n Substituting from eq. (p4.4-7) into (p4.4-5), k e can be determined. 22

23 So, add a To workspace component in the testntf model. Make double click on it in order to name it x and set the Maximum number of Timesteps parameter equal to as in u and y workspace components. Then connect it to the Sine Wave component, and perform simulation. Then in MATLAB window compute first ue ( n) and then k e by entering the following commands: ue=u-mean(u.*x)/mean(x.*x)*x; ke=mean(y.*ue)/mean(ue.*ue) ke =.547 From the figure p4.4-, it is clear that when this value of gain more accurate estimation of the quantization noise shaping is achieved. k e is taken into consideration, a To obtain the figure p4.4-, the modulator has to be simulated for a long time in order to estimate the power spectral density more accurately by computing the average spectrum. So, in the testntf window open the Simulink control panel window by selecting the Parameters command from the Simulation menu, as it has been described in Step 3 of the problem 4.3, and set in the Stop Time parameter instead of Then by double clicking in the y workspace component, insert the same number. After these changes, create the following Script file: 23

24 nw=256; n=48576/nw; % Separate the total number of data points to nw windows % Find the average spectrum w=blackman(n); spec=zeros(n,); % Window for i=:nw wy =y(n*(i-)+:n*i).*w; cnorm= mean(y.*y)/ mean(wy.^2); spec=cnorm*(abs(fft(wy/n)).^2); spec=spec+spec; end spec=spec/nw; % Normalization of Signal Power % Average spectrum % Plot modulator output spectrum f=[:n/2]/n; figure(); plot(f,*log(spec(:n/2)),'w') axis([ ]); hold on % ---- Plot the estimation of spectrum using the Inverse Chebyshev Transfer Function [z,p,k]=cheby2(5,,/64,'high'); [NTFnum,NTFden]=zp2tf(z,p,); [H,wf]=freqz(NTFnum,NTFden,496); % Find the magnitude response px=(.45^2)/2 cnorm=(-px)/mean(abs(h).^2)/n; % Normalization of the magnitude response plot(wf/pi/2,*log(cnorm*abs(h).^2),'w') % ---- Plot the estimation of spectrum using the transformed Inverse Chebyshev ke=.547 NTFdenk=ke*NTFden+(-ke)*NTFnum; % New denominator [Hk,wf]=freqz(NTFnum,NTFdenk,496); % Find the magnitude response cnorm2=(-px)/mean(abs(hk).^2)/n; % Normalization of the magnitude response plot(wf/pi/2,*log(cnorm2*abs(hk).^2),'w') In order to zoom in in-band, execute the same commands using nw=6, making also the proper changes in the plot commands. 24

25 4.5 Simulate a Σ modulator whose NTF is one of the Inverse Chebyshev functions in Table 4.4. Repeat with the same NTF having the same poles and zeros at dc. Verify that the gain in SNR is as given in Table 4.3. Answer: Again, we shall examine the same modulator as in the previous problems. Change all the zeros to be at dc, and compute the new coefficients of the filter Lz (). Next insert them in the Filter component of the testntf model and save the changes. Then execute the script file findsnr which you created in subquestion c) of problem 4.3 and draw the plot by entering plot(amp,snr) hold on After that load the data file which contains the SNR values of the modulator before changing the position of the zeros and also draw their plot by entering load data plot(amp,snr) The following SNR diagram is obtained 2 SNR 8 Zeros of Inverse Chebyshev Transfer Function All zeros at dc Amplitude () Figure p4.5-. SNR as a function of the input amplitude. It can be verified that when all zeros are at dc, the SNR plot is decreased almost 7 as it is estimated theoretically by means of Table