Homework 6 Solutions

Transcription

1 8-290 Signals and Systems Profs. Byron Yu and Pulkit Grover Fall 208 Homework 6 Solutions. Part One. (2 points) Consider an LTI system with impulse response h(t) e αt u(t), (a) Compute the frequency response H(jω) of the LTI system. (b) What is the output y(t) of the system when input is e jω 0t? (c) What is the output y(t) of the system when input is x(t) cos(ω 0 t)? (Hint: Use Euler s identity and the frequency response of the system.) (d) Compute the output y(t) of the system when input is x(t) 2 cos 2 (πt). Solution: (a) Compute the frequency response H(jω) of the LTI system. H(jω) 0 e ατ u(τ)e jωτ dτ e (α+jω)τ dτ α + jω e(α+jω)τ 0 α + jω (b) What is the output y(t) of the system when input is e jω 0t. y(t) h(t) e jω 0t h(τ)e jω0(t τ)dτ e jω 0t h(τ)e jω 0τ dτ e jω 0t H(jω 0 ) (c) What is the output y(t) of the system when input is x(t) sin(ωt). (Hint: Use the Euler s ind identity and the frequency response of the system). When the input is x(t) cos(ω 0 t), we can write the input as x(t) cos(ω 0 t) ejω 0t + e jω 0t 2

2 2 Homework 6 Solutions The output of the system to inputs e jω 0t and e jω 0t are H(jω 0 )e jω 0t and H( jω 0 )e jω 0t. So, we can have the output y(t) as follows: y(t) 2 H(jω 0)e jω 0t + 2 H( jω 0)e jω 0t 2(α + jω 0 ) ejω 0t + 2(α jω 0 ) e jω 0t (α jω 0 ) 2(α + jω 0 )(α jω 0 ) ejω 0t (α + jω 0 ) + 2(α + jω 0 )(α jω 0 ) e jω 0t (α jω 0)e jω 0t + (α + jω 0 )e jω 0t 2(α 2 + ω 2 0) α(ejω 0t + e jω 0t ) + jω 0 (e jω 0t e jω 0t ) 2(α 2 + ω 2 0) 2α cos(ω 0t) 2jω 0 (j sin(ω 0 t)) 2(α + ω 2 0) α cos(ω 0t) + ω 0 sin(ω 0 t) (α 2 + ω 2 0) (d) Compute the output y(t) of the system when input is x(t) 2 cos 2 (πt). When the input is x(t) 2cos 2 (πt), we can write the input as x(t) 2 cos 2 (πt) cos(2πt) Apply the conclusion in Question (c), we have have the output y(t) as follows: y(t) α cos(2πt) + 2π sin(2πt) (α 2 + 4π 2 )

3 Homework 6 Solutions 3 2. (9 points) Consider the impulse response given below. 2 h(t) t (a) Compute the frequency response H(jω) of the system. (b) Consider a system with impulse response h (t) h(t 3). Compute the frequency response H (jω), and represent H (jω) in terms of H(jω). (c) Consider a system with the impulse response give below. Compute the frequency response H 2 (jω) of the system and represent it in terms of both H(jω) and H (jω). 2 h 2 (t) t Solution:

4 4 Homework 6 Solutions (a) Compute the frequency response H(jω) of the system. H(jω) 2 0 jω 2 0 h(t)e jωt dt (u(t) u(t 2))e jωt dt e jωt dt e j2ω jω (b) Consider a system with impulse response h (t) h(t 3). Compute the frequency response H (jω), and represent H (jω) in terms of H(jω). H (jω) 2 3 jω 5 3 h (t)e jωt dt (u(t 3) u(t 5))e jωt dt e jωt dt e j3ω e j5ω jω e j2ω jω e j3ω H(jω) e j3ω Thus we have H (jω) e j3ω H(jω). (c) Consider a system with the impulse response give below. Compute the frequency response H 2 (jω) of the system and represent it in terms of H(jω), and both

5 Homework 6 Solutions 5 H(jω) and H (jω) H (jω) 2 0 h (t)e jωt dt (u(t) u(t 2) u(t 3) + u(t 5))e jωt dt e jωt dt jω jω 5 3 e j2ω jω 5 e j2ω jω H(jω) H (jω) 3 e jωt dt + ej3ω + e j5ω jω ej3ω e j5ω jω H(jω) e j3ω H(jω)

6 6 Homework 6 Solutions 3. (9 points) A continuous time system H has the frequency response H(jω) (a) Find the magnitude response and plot it. (b) Find the phase response and plot it. 4π 4π + jω. (c) Using H(jω), find the output y(t) for the input x(t) 4 cos(4πt) + 4 cos(2πt). Solution: (a) Magnitude response, H(jω) H(jω)H (jω) 4π ( 4π + jω )( 4π 4π jω ) (4π) 2 (4π) 2 + ω 2 4π 6π2 + ω 2 H(jω) ω (b) Phase response, H(jω) H(4π) H(4π + jω) 0 tan ( ω 4π ) tan ( ω 4π )

7 Homework 6 Solutions 7 H(jω) ω (c) For input x(t) 4 cos(4πt) + 4 cos(2πt), we have x(t) 2e 4jπt + 2e 4jπt + 2e 2jπt + 2e 2jπt So, we have the output of x(t) given as follow: y(t) 2H(4jπ)e 4jπt + 2H( 4jπ)e 4jπt + 2H(2jπ)e 2jπt + 2H( 2jπ)e 2jπt 4π 4π 4π 4π 2 4π + j4π e4jπt + 2 4π j4π e 4jπt + 2 4π + j2π e2jπt + 2 4π j2π e 2jπt 2 ( j)e4jπt + ( + j)e 4jπt ( j)( + j) 2 (e4jπt + e 4jπt ) j(e 4jπt e 4jπt ) ( 3j)e2jπt + ( + 3j)e 2jπt ( 3j)( + 3j) 2 cos(4πt) + 2 sin(4πt) cos(2πt) sin(2πt) + 2 (e2jπt + e 2jπt ) 3j(e 2jπt e 2jπt ) 5

8 8 Homework 6 Solutions 4. (2 points) Consider a LTI system with impulse response h[n] a n u[n], where a <. (a) Determine the frequency response of the system. (b) Find the magnitude response and plot it, given a 2. (c) Find the phase response and plot it, given a 2. (d) Consider a LTI system whose impulse response h [n] is a time-shifted version of h[n], i.e., h [n] h[n n 0 ]. Compute the frequency response H (e jω ), and represent H (e jω ) in terms of H(e jω ). (e) Given n 0 2 and a, plot the magnitude response and phase response of the 2 system described in (d). Solution: (a) h[n] a n u[n] Assume we a <, otherwise the result does not converge. (b) H(e jω ) a n u[n]e jωn a n e jωn n0 (ae jω ) n n0 ae jω H(e jω ) ae jω a cos( Ω) ja sin( Ω) a cos(ω) + ja sin(ω) + a2 2a cos(ω)

9 Homework 6 Solutions 9 2 H(e j ).8.6 H(e j ) (c) H(jΩ) () ( a cos(ω) + ja sin(ω)) ( ) a sin Ω 0 tan a cos(ω) ( ) a sin Ω tan a cos(ω)

10 0 Homework 6 Solutions 0.6 H(e j ) H(e j ) (d) a n n 0 u[n n 0 ] Assume we have a <, otherwise the result does not converge. H(jΩ) a n n 0 u[n n 0 ]e jωn e jωn 0 e jωn 0 ae jω nn 0 (ae jω ) n n0

11 Homework 6 Solutions (e) H(jΩ) e jωn 0 ae jω e jωn 0 ae jω ae jω 2a cos(ω) 2 H(e j ).8.6 H(e j ) H(jΩ) e jωn 0 ( a cos(ω) + ja sin(ω)) ( ) a sin Ω Ωn 0 + tan a cos(ω)

12 2 Homework 6 Solutions 8 H(e j ) 6 4 H(e j )

13 Homework 6 Solutions 3 Part Two 5. (2 points) Let h[n] u[n] be the impulse response of an LTI system. Let x[n] 3 n cos ( π n) be the input to the system. 4 (a) Evaluate the output y[n] h[n] x[n]. Simplify your answer so that there are no complex exponentials. (b) Express x[n] as a sum of complex sinusoids, i.e., compute a k such that x[n] k a ke jkω0n, where Ω 0 π. (Hint: Use Euler s identity) 4 (c) Evaluate the frequency response H ( e jω) of the system with impulse response h[n]. (d) Compute the output y[n] k a kh ( e jkω 0) e jkω 0 n. Simplify your answer so that there are no complex exponentials. How does your answer compare to y[n] computed in part (a)? Solution:

14 4 Homework 6 Solutions (a) y[n] x[k]h[n k] n 3 n n ( π ) cos 4 k u[n k] 3n k ( π ) cos 4 k 3 n k ( π ) cos 4 k 3 k n 3 n 2 (e π 4 k + e π 4 k )3 k ( n ) n (3e j π 2 3 n 4 ) k + (3e jπ 4 ) k 3n e j π 4 n 2 3 n + 3n e j π 4 n 3e j π 4 3e jπ 4 ( ) 3 e j π 2 3 e j π 4 n 3 + e j π 4 3 e j π 4 n 4 ( ) j e j π 4 n j e j π 4 n e j π 4 n j 2 + 3e j π 4 n 6 2 j π 2 (ej 4 n + e j π 4 n 3 2 ) j 40 2 π 2 (ej 4 n e j π 4 n ) ( π ) cos 4 n 3 2 ( π ) j j sin 4 n ( π ) cos 82 4 n ( π ) sin 82 4 n Acceptable solution (or similar): ( e j π 4 ) e j π 4 n 3 + e j π 3 e j π 4 n 4

15 Homework 6 Solutions 5 (b) ( π ) x[n] cos 4 n 2 (ej π 4 n + e j π 4 n ) a 2 a 2 a k 0 otherwise (c) (d) y[n] k H(e jω ) h[n]e jωn 0 3 n u[n]e jωn 3 n e jωn ( e jωn 0 a k H(e jω 0k )e jω 0n 3 e jωn e jω ) n a H(e jω0( ) )e jkω0( )n + a H(e jω0() )e jkω 0()n 3 e j π 2 3 e j π 4 n + 3 e j π e j π 4 n ( π ) cos 82 4 n ( π ) sin 82 4 n Acceptable solution (or similar): ( ) 3 e j π 2 3 e j π 4 n 3 + e j π 4 3 e j π 4 n 4

16 6 Homework 6 Solutions 6. (5 points) MATLAB. When we record signals in the real world we often get a combination of the desired signal and unwanted noise. In this problem we will try to clean a noisy signal with filters. Please download hw6 matlab.zip from the course website ( zip). (a) Load the data in hw6 matlab.mat, which contains the following: i. two signals sig orig and sig noisy, ii. the impulse response of three filters IR, IR2, IR3, iii. sampling frequency fs. Use the function view frequencies as you did in Homework 5 to plot the frequency content of the two signals sig orig and sig noisy. Submit the following: Plots of the frequency components of sig orig and sig noisy and your code for generating them. Briefly describe the difference between the two plots. Based on the plots for sig orig and sig noisy, what type of filter should we use to remove the noise? (b) Convolve the signal sig noisy with each of the three filters and plot the frequency content of the outputs. You can use soundsc(sig, fs) to play back a signal sig. Submit the following: Solution: Plots of the frequency components of the three filtered outputs. Your code for generating the plots. Listen to your outputs and briefly describe how well each filter denoises the signal based on what you hear and the plots you generated. (a) The original signal has one spike at around 220 Hz while the noisy signal has a lot of small spikes at different frequencies besides the main spike. A band-pass filter should be used. load (' hw6_matlab.mat ') [f,amp ] view_frequencies ( sig_orig,fs ); plot (f,amp ) title (' Original Signal ') xlabel (' Frequencies (Hz )') ylabel (' Magnitude ') [f,amp ] view_frequencies ( sig_noisy,fs ); plot (f,amp )

17 Homework 6 Solutions 7 title (' Noisy Signal ') xlabel (' Frequencies (Hz )') ylabel (' Magnitude ') (b) IR2 works the best because it s a band-pass filter centered around the frequency

18 8 Homework 6 Solutions of the major spike we would like to keep in the original signal, removing a lot of the unwanted frequency components at each side. IR and IR3 can do some denoising and maintain the major spike, but they still keep a lot of the unwanted frequency components due to their low-pass or high-pass nature. sig conv ( sig_noisy,ir,'same '); sig2 conv ( sig_noisy,ir2,'same '); sig3 conv ( sig_noisy,ir3,'same '); [f,amp ] view_frequencies (sig,fs ); plot (f,amp ) title (' Convolution with IR ') xlabel (' Frequencies (Hz )') ylabel (' Magnitude ') [f,amp ] view_frequencies (sig2,fs ); plot (f,amp ) title (' Convolution with IR ') xlabel (' Frequencies (Hz )') ylabel (' Magnitude ') [f,amp ] view_frequencies (sig3,fs ); plot (f,amp ) title (' Convolution with IR ') xlabel (' Frequencies (Hz )') ylabel (' Magnitude ')

19 Homework 6 Solutions 9

20 20 Homework 6 Solutions

21 Homework 6 Solutions 2 7. (20 points) MATLAB. We have evaluated filtering to blur images in the Homework 5. Now, we will get a closer look about low-pass, high-pass and band-pass filtering of images in this problem. Recall that convolution for images was defined as J(x, y) W U w W u U I(x u, y w)h(u, w) where I(x, y) was the image and h(x, y) was a (Gaussian) blur impulse response. We can instead have impulse responses which enable detection of edges in images. (a) (6 points) Before we handle images, consider the impulse response h[n] δ[n] δ[n ]. Compute (without using Matlab) the magnitude of its frequency response, H(e jω ). Is this a low pass, high pass or band pass filter? Justify your answer. Submit the following: Magnitude response Filter type Justification for the type of filter. (b) (3 points) Now load the Matlab file, hw6 matlab2.mat. You will find an image test image, and the impulse response of four 2 2 filters f, f2, f3, f4. View the image from the Matlab workspace: imshow(test_image, []); Type the following in your workspace next: disp(f); % try f2, f3 and f4 and do this for all four filters. What types of filters are they? (Hint: Answer this question by referring to your solutions to part (a)). Submit the following: Type of filters f, f2, f3, f4 low pass, high pass or bandpass with one line justification for each filter. (c) (5 points) Next, apply each filter f, f2, f3, f4 to test image separately and evaluate the filtering property by plotting the original image and output as follows: image conv2(test_image, f, 'same'); imshow([image], []);

22 22 Homework 6 Solutions Is the output for each filter consistent with your answer to previous problem? What is the difference between filter f and the other filters? Submit the following: Images showing the original test image and the outputs of each filter, along with explanation of the outputs. (d) (6 points) For which filters do the outputs show distinct edges? For those filters, what is the difference among their outputs? Based on your answers, do edges in images represent high frequencies or low frequencies? Submit the following: A list of which filters show edges in their outputs and a short description of the difference between those outputs. Do edges in images represent high frequencies or low frequencies? your answer. Justify Solution: (a) H(e jω ) n n h[n]e jωn (δ[n] δ[n ])e jωn e jω(0) e jω() e jω cos (Ω) + j sin (Ω) H(e jω ) 2 ( cos (Ω)) 2 + (sin (Ω)) 2 + cos 2 (Ω) 2 cos(ω) + sin 2 (Ω) 2( cos(ω)). Notice that H(e j0 ) 0, H(e jπ ) 4. Hence this is a high pass filter (b) By plotting the impulse response of each filter entry, we get, f is low pass filter and f2, f3, f4 are high pass filters. Or simply based on the value of filter entries, we can get the same result. (c) f is visibly a low pass filter and f2, f3, f4 are high pass filters. Figures of each output show as follows:

23 Homework 6 Solutions 23 Figure : Original Test image. Figure 2: Output of first filter f.

24 24 Homework 6 Solutions Figure 3: Output of second filter f2. Figure 4: Output of third filter f3.

25 Homework 6 Solutions 25 Figure 5: Output of fourth filter f4. (d) () We have seen f2, f3, f4 show distinct edges. f2 shows vertical edges, f3 shows horizontal and f4 shows diagonal edges. (2) As we seen in the figure 2,3 and 4, distinct edges comes up because the high filters have filter out the original signals at the edge. Therefore, edges represent high frequencies. Furthermore, we can also see edges has a higher rate of change of intensity per pixel in the image processing context.