Assignment 2. Signal Processing and Speech Communication Lab. Graz University of Technology

Transcription

1 Signal Processing and Speech Communication Lab. Graz University of Technology Assignment 2 This homework has to be submitted via to the address hw1.spsc@tugraz.at not later than Let the subject of the be YourMatro YourColleague smatro. The body of the should be empty (nobody will read it). A complete project consists of Matlab/Octave files (*.m) and a simulation protocol in PDF format. You have to zip all these files to a single file with name YourMatro YourColleague s Matro.zip which has to be attached to the . In addition to the , you have to throw your printed (paper format) simulation protocols and your analytic solutions into our mailbox at Inffeldgasse 16c, ground floor, not later than (note that you cannot access the mailbox on weekends). For each problem, staple your solutions separately. Use the print-out of the respective problem assignment as the title page(s). Don t forget to fill in your name(s), matr. number(s), and group number(s). If you typeset your analytic solutions with L A TEX, you can get bonus points. If you submit a handwritten protocol, be aware that it should be clear, tidy and readable. Otherwise, you will be charged with five penalty points.

2 Analytical Problem 2.1 (4 points) Let x[n] = (1 + e j 5π 1 2 )δ[n] ( 2 + π ej 2 )δ[n 1] denote a complex-valued discrete-time signal with DTFT X ( e jθ). (a) [2 point(s)] Using properties of the Fourier transform calculate the following expressions: 1. X ( e jθ) θ=0 2. +π π X ( e jθ) 2 dθ (b) [1 point(s)] Using again properties of the Fourier transform calculate the time domain signals following the expressions below: 1. y 1 [n] = DTFT 1 {I{X ( e jθ) }} 2. y 2 [n] = DTFT 1 {X ( e jθ) } (c) [1 point(s)] Sketch the real and imaginary parts of y 1 [n] and y 2 [n] separately.

3 Analytical Problem 2.2 (6 points) The following impulse response of a system is given: h 1 [n] = u[n] u[n 2] u[n 3] + u[n 5], where u[n] is the discrete-time unit-step function. (a) [1 point(s)] Sketch the impulse response h 1 [n]. What property does the phase-response of this system have (justify your answer)? (b) [2 point(s)] Calculate H 1 (e jθ ) = DTFT{h 1 [n]} without using DTFT-pairs and sketch its amplitude and phase spectrum for 2π < θ < 2π. Make sure that the zeros of the amplitude spectrum are correctly aligned on the frequency axis. ( (c) [1 point(s)] Find a scaling factor α so that α H ) 1 e jθ θ= π 2 sequence h 1 [n] = DTFT 1 {αh 1 ( e jθ ) } is equal to one. Determine the (d) [2 point(s)] Consider a second system with impulse response ( ) 2π h 2 [n] = cos 3 n. h 1 [n] and h 2 [n] are connected in series yielding the overall impulse response h 3 [n] given by: h 3 [n] = h 1 [n] h 2 [n] Determine h 3 [n]. (Hint: Illustrating the frequency responses in Octave may help.)

4 Analytical Problem 2.3 (5 points) Consider the following discrete-time sequence x[n] = δ[n] δ[n 1] + δ[n 2] (a) [1 point(s)] Find the sequence x 1 [n] = DTFT 1 {X ( e jθ) e jθn 0 }, where X ( e jθ) is the DTFT of x[n] and n 0 is an integer. (b) [1.5 point(s)] ow let X[k] be samples of X ( e jθ), i.e.: X[k] = X ( e jθ) θ= 2πk, k = 0, 1,..., 1, with = 5. Find the -point sequence x 1 [n] that corresponds to: x 1 [n] = DFT 1 2πk {X[k]ej n 0 }. (c) [1.5 point(s)] Sketch x 1 [n] and x 1 [n] for n = 0, 1,..., 1 for n 0 = 3. Can you find a value n 0 3 that yields the same x 1 [n]? (d) [1 point(s)] What is the minimum DFT-length so that x 1 [n] = x 1 [n] holds for n = 0, 1,..., 1 if n 0 = 3? Justify your choice.

5 Octave Problem 2.4 (10 points) Consider the following 1 -point sequence representing a discrete-time domain signal x 1 [n] = A 1 cos (ω 1 n + φ 1 ), n = 0,..., 1 1, where 1 is some fixed integer and the normalized angular frequency ω 1 is defined by ω 1 = 2π f 1 f S. Further, f 1 is a frequency in (Hz) and f S is the sampling frequency. (a) [1 point(s)] Let s assume we have the following parameters: A 1 = 2, f 1 = 7 4 Hz, f S = 8Hz, φ 1 = 0. Given the sampling frequency f S choose 1 so that it corresponds to 4 s. ow generate the index sequence {n 1 } = {0, 1,..., 1 1}, which can be expressed as a vector n 1 in Octave. Given the parameters defined above compute the signal vector x 1 and visualize it using figure and stem. Make sure that the x-axis is indexed correctly (with {n 1 }) and don t forget to label both axes! (b) [2 point(s)] Compute the 1 -point DFT of x 1 [n] X 1 [k] = DFT{x 1 [n]} = 1 1 n=0 x 1 [n]e j2πkn/ 1 using the function fft. X 1 [k] is a complex spectrum consisting of 1 complex values. Illustrate the absolute value of the spectrum by applying abs, figure and stem. Again make sure that the x-axis is indexed correctly. For this purpose generate the frequency index sequence k 1 : {k 1 } = {0, 1,..., 1 1} What can you say about the absolute values X 1 [k] at k = f 1 1 f S and k = 1 f 1 1 f S (the maxima of X 1 [k] )? Can you compute A 1 from these values? If the underlying discrete-time sequence x[n] is real-valued, is there in general a way to obtain the angle of X[k] at k = /2+a if one only knows the angle of X[k] at k = /2 a (with integer a {0,..., /2})? (c) [2 point(s)] ow extend your vector x 1 by 0 = 2000 zeros with the help of the function zeros (this operation is a general concept called zero-padding). We will refer to the new vector as x 1. Evaluate the new vector s length Ñ1 by utilizing one of the following functions: numel, length or size. Compute the corresponding Ñ1-point DFT X 1 [k] with fft. The normalized angular frequency at frequency index k is given by θ k = 2πk. Compute the values of θ k for the DFT lengths 1 and Ñ1 separately to obtain θ k,1 and θ k,1, respectively. In order to illustrate the impact of the zero-padding we will now compare the DFT representations of x 1 [n] and x 1 [n]. Use plot to plot X 1 [k] against θ k,1 and stem to visualize X 1 [k] against

6 θ k,1, both in the same plot. (Hint: hold on lets you overlap the two curves.) What do you observe? How are the DTFT and the DFT of a sequence related? (d) [2 point(s)] We now define a second signal with the following parameters: A 2 = 2, f 2 = 8 5 Hz, f S = 8Hz, φ 2 = 0, 2 = 1. In Octave, generate vectors representing an index sequence n 2 and the corresponding discretetime sequence x 2 [n] following and {n 2 } = {0, 1,..., 2 1}, x 2 [n] = A 2 cos (ω 2 n + φ 2 ), n = 0,..., 2 1, with ω 2 = 2π f 2 f S. ow follow the same steps as in task 2.4 (b) in order to illustrate the 2 -point DFT X 2 [k] of x 2 [n] by figure and stem (don t forget to compute the correct index vector k 2 to have a valid representation!). What do you observe? ow zero-pad the vector x 2 with 0 = 2000 zeros in order to obtain x 2 with Ñ2 entries. Compute the Ñ2-point DFT X 2 [k] of x 2 [n] and generate two vectors containing the normalized angular frequencies θ k,2 and θ k,2 for both DFT lengths, 2 and Ñ2. Similar to 2.4 (b) illustrate the absolute value of both spectra in one plot using figure and plot for plotting X 2 [k] against θ k,2 and stem to plot X 2 [k] against θ k,2 (both in one plot again). Discuss your observations. (e) [1 point(s)] Find a value for 2 yielding X 2 [k] to be nonzero only for the two frequency indices k = f 2 2 f S and k = 2 f 2 2 f S i.e.: { 2 A 2, k = f f X 2 [k] = S, 2 f 2 2 f S 0, elsewhere Illustrate your results by plotting X 2 [k], resulting from using the new 2 you have chosen. (f) [1 point(s)] By simply restricting the time-domain signal to the interval n = 0, 1,..., 1 we window the signal with a rectangular window of length. The window choice plays an important role in signal-processing applications and the rectangular window is not always the optimal choice. Thus, in this task we apply a blackman window on a discrete-time signal and compare it to the outcome of a rectangular window. Generate the signal ( ) 5ω1 x 3 [n] = A 1 cos 4 n, with the same vector n 1 and parameters A 1 and ω 1 as in 2.4 (a). Compute the signal x 4 [n] following x 4 [n] = x 3 [n] + x 1 [n],

7 represented by the vector x 4. Generate a blackman-window as a vector w B of length 1 by using blackman(), with the window-length as argument. The rectangular window vector w R of same length can be obtained by using ones. The windowed vectors can now be computed by x 4,R = x 4 w R x 4,B = x 4 w B, where the operator denotes the Hadamard product, which means that the vectors are multiplied elementwise. Zero-pad both vectors with 0 zeros and plot the absolute values of the corresponding spectra via fft, abs, plot and hold on in one figure. Label the x-axis correctly and use legend in order to clarify which curve belongs to the rectangular window and which one to the blackman window. What do you observe? (g) [1 point(s)] ow zero-pad the window vectors w R and w B with 0 zeros to obtain w R and w B, respectively. Compute their DFTs W R and W B, by using fft. ow normalize (divide) both vectors by the sum (sum) of the absolute values of their entries individually and you get W R,norm and W B,norm. Plot both, W R,norm and W B,norm, into the same figure. (Hint: Here, the command fftshift may help; you can obtain the corresponding frequency-axis by θ k,1 π). Do not forget to label the x-axis correctly and use a legend. What is a possible disadvantage of the rectangular window (what can you tell about the main lobes and the side lobes of the two window types)?