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ECE114_2005FALL_EXAM2__[0]

# ECE114_2005FALL_EXAM2__[0] - The Cooper Union Department of...

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Unformatted text preview: The Cooper Union Department of Electrical Engineering ECE114 Digital Signal Processing 155me . Deoemberl,2{}05 Time: 90 min. Closed book, closed notes. No calculators. 1. [12 pts.) The output of a D/A mnverter is modeled as: 2 {cup (t — RT) (1) 73:-“ where {on} are the data sample values and p (t) is an underlying pulse shape. Here, T =_1/f, where f, is the sampling rate in Hz. Let X (w) denote the DTFT of .1:(n); let X ( f) denote an analog spectrum (with f in Hertz) related to X (to) via: X (f) = X (”Nu-skiff. (2) Also let. P (f) = F{p (15)}, the OTFT ofp(t). Two basic results are: :r{ 2 we we} = \$120) (3) where the 5 (-) is of the continuous type, and: 5F{ 2 watt—m} =§Xum (4) (a) 1? (f) is periodic with period ,2; brieﬂy Show how this is forced by (2). (1)) Given (3) (do not attempt to prove it), pmve (4); this should require no more than a. few words. (c) Over what range of frequencies does the behavior of |P (f)| determine the imaging distortion caused by the D/A conversion process? Brieﬂy explain this by refer- encing (4); that is, do not just state the result, but show how it is apparent from this equation. (d) Over what range of frequencies does the behavior of IP (f )I determine the am- plitude distortion caused by the D [A conversion process? Again, reference (4) to explain it. 2. [4 pts.] The modiﬁed Welch periodogrsrn is to be computed for a block of data. The key parameters are: 512-point DFT computed on 100 frames with 256 point overlap. (a) What is the total length of the data block that is required? (1)) What is the appropriate length of the window function that should be employed? (0) If it is desired to improve the spectral resolution, changing which of the following will have the most direct effect? The size of the DFT (512), the number of frames (100) or the amount of overlap [256)? -"(d) If it is desired to reduce the variance of the periodogram, changing which of the following will have the most direct effect? The size of the DFT (512), the number of frames (100) or the amount of overlap (256)? 3. [10 pts.] A (causal, real) linear-phase FIR ﬁlter of length 6 is partially speciﬁed by: M0) = 2, h(1) = 3, 11(2) =5 (a) Are the other coefﬁcients completely determined? If not, how many choices are there? Give all possible solutions. If one or more coefﬁcients are completely free (no constraints at all), say 30. (b) Repeat for the case of an FIR ﬁlter of length 7. (0) Of all the different cases, lengths [i and 7, which could perhaps be used for a digital realization of a Hilbert trmisﬁirmer'? (d) For the case of length 6 only, pick one of the possible solutions. Drew's. ﬁlter structure that not only has a minimum number of delays but also has a minimum number of multipliers. (e) Does your realization require more, the same or fewer adders as a transversal ﬁlter? 4. [4 [3123.] Let h“ (n) of length N be an “unwindowed” preliminary design of a linear— phase FIR ﬁlter. A window to (n) is to be applied to result in the ﬁnal design h (n). (a) Write the formula that is used to compute h from h." and w? (b) What are the lengths of h and w? [2 pts.] A linear-phase Remez FER. ﬁlter is to be designed to have gain in the range 1 :l: 0.1 in the passband and maidnmm gain 0.01 in the stopband. Suppose a weight value W = 1 is speciﬁed in the passband. .01 (a) What is the weight in the stopband? (b) Suppose the CAD tool that designs the ﬁlter “demands" that a weight be speciﬁed in the transition (don’t care) bands. What value should be used? 6. [8 pts.] Write the DFT and IDFT formulas in terms of complex exponentials, and again expressed in terms of the twiddle factor WN. 'i'. [2 pts.] What is the value of the twiddle factor W”? 2 8 [5 pts.] Given the four-point signals: a: = {2,1, —3,4} *5 {3: 2: 5'.- 7.} We adopt the convention of padding with zeros, when necessary. Let y.‘ (n) denote the 4-point circular convolution, and y; (rt) the linear convolution. (a) Compute the four—point circular convolution ya in) for n = I only. (13) What is the support of ”(11)? (c) Can we compute yr (1) from m, (n)? Ifso, how? 9.‘ [5 pts.] For the signal 1: in the previous problem, compute the 8-point DFT for index k = 5: X3 (5). You must simplify to the form (a + inf?) +j (c+ (ix/27 Where a,b,c,d are rational. You will not get full credit unless you perform the simpliﬁcation! 10. [10 pts.] A digital spectrum analyzer samples a signal at a rate 20kHz, collects a block of 1000 points, and performs a DFT. Note that we are not using a power-of-2 DFT here to make the numbers come cut “nicer”. (a) What is the bin spacing, in Hertz? (b) A peak is oNerved corresponding to DFT index k = 5. What is the associated frequency in Hertz? (c) Assuming the sampled signal is reel, at what other DFT index it will a peak be observed? ((1) The MATLAB function ﬂ'tshift swaps the two halves of a vector; for example, ﬁshtftﬂa b c dj) returns [a d a b]. If we apply ﬂ‘tahift to the vector of DFT coefﬁcients, then the associated frequency ooorcinates (on a. Hertz scale) can he created via fa : Af : 1', where f0 and f; are the ﬁrst and last frequencies, and Af is the step-size. Specify ﬁg, 3", and Af. (e) Referring to the previous pert, whet negative frequency coordinate is closest to 0? What is the corresponding DFT index k (k 2: 0)? ill}. [10 pts.] A real sinewave at 2MHz is sampled at 013MHz. The data samples are buffered and fed to a D/A converter operating at 5 Hz. Assume the A/D and D/A converters employ the standard anti-aliasing and anti imaging ﬁlters. Neglect issues of butter underﬂoor/overﬂow caused by the diﬁerentsa pling rat-I. (a) What are the cutoff frequenciw of the anti-all and anti-imaging ﬁlters? (h) Specify the frequency of the ZMHz input to th 10M H z A/D converter on the following scales: normalized digital radian freque cy, as a fraction of the sampling rate, as a. fraction of the Nyquist bandeddth. (c) [f not for the anti-aliasing and anti-imaging ﬁlt , list all (positive) frequencies up to 30M H 2 that would alias into 2M H .2 at the A D converter, and all frequencies up to SOME: that would emerge at the output f the D/A converter. (d) W’ith the proper ﬁlters in place, what [positive] uency (or frequencies) appear at the output of the D/A converter? 12.‘ [5 pts.] Consider the design of a digital IlIt ﬁl based on an analog prototype employing either impulse invariance or bilinear tr rm methods. For each method, specify which of the following properties are preser . No explanation- just set up a table and check off all the one; that apply. - (a) Equiripple paesband gain. (b) Munotonic stopband gain. (c) Maxﬂat passband. (:1) Rise time. (e) Stability. . 13. [2 pts.] TRUE or FALSE (no explanation): In the co ventional terminology employed in the literature, a. “digital elliptic ﬁlter" could ha been designed by applying the impulse invariance method on an analog elliptic pro type ﬁlter. 14. [2 ptﬂ.] TRUE or FALSE (no explanation): In the impulse hivariance method, the polesandzerosoftheanalogﬁlteraremeppedaccor ' gtothemlez=e‘r. ' '15; [2 pts.l What is the mapping .5 = F (z) that is cm oyod in the case of the bilinear - transform design method? Assume prewarping is 16. " [8 pts.] A digital bandpass Chebyshev I ﬁlter operating at a sampling rate of 1M H z is to meet the following speciﬁcations: o 2:13 variation in the passband from 200kHz_ to 300kHz; a 3MB attenuation in the stopband from 150k}! 7. to 350kHz. (a) Sketch the speciﬁcations with the frequency,r ands labeled in Hertz, and again in normalized digital radian frequency. .{b} Sketch the speciﬁcations of the prototype analog bandpass ﬁlter, with pmband edges labels [2,1, 91,2 and stopband edges labeled 9.1, (132 (do not compute these valuesl). If as denotes the digital frequency speciﬁcations in normalized radian units, and (3’s are in units rad/ see, what is the formula that should be used to compute the (2's from the w’s? (c) The speciﬁcations of a lowpass ﬁlter (2‘? are computed from the bandparﬁ Q's via. the formula: as - 92 39 Write explicit formulas for the parameters 00,8 in terms of the bandpass Q’s. (d) What frequencies on the lowpass scale, 51“”, do the passband edges 0P1, \$1,; map to? You should know the answer without actually computing it. 9‘? = (9.) Suppose you are given that the stopband edges 9,1,954 map to lowpass values [1‘9 = 3.3511 and 4.0762, respectively. Which of their: two values should be used to design the lowpass prototype ﬁlter? (f) Suppose the ‘ﬁ‘aw” formula for the required order of the lowpass prototype returns the value 11 = 2.3489. What is the order of the lowpass analog prototype? The bandpass analog prototype? The final digital ﬁlter? Here order refers to the number of poles of the system. 17. [4 pts.] Characterize the pamhand and stopbaud of each of the following as either monotonic or equiripple: (a) Butterworth (b) Chebyshev I (c) Chebyshev [I (d) elliptic Figure 1: Magnitude pronse of a Window Function 18. [5 pts.] Figure 1 shows the magnitude response of a window function employed by a digital spectrum analyzer. {a} Determine the nulLto—uull mainlobe width and the peak sidelohe level. Clearly Show how you determined these values (draw a quick sketch of the curve and label it to Show how you got these values). (12-) Which 05E these two parameters is generally associated with spectral resolution? [0) Which is generally associated with spectral leakage? (d) This window function is replaced by a. Chebyshev window, exhibiting sidelobee with equal amplitude; this Chebyshev window has the same peak sidelobe level as the indicated window ﬁmction. Will the Chebyshev window have a. larger or smaller null-to-null mainlobe width? ...
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