2 May 2000 Name: EE 438 Final Exam Spring 2000 You have 120 minutes to work the following six problems. Each problem is worth 25 points. Be sure to show all your work to obtain full credit. The exam is closed book and closed notes. Calculators are permitted. 1. (25 pts.) Consider the system shown in the following diagram which operates at a sampling frequency of f s Hz x(t) A/D x[n] y[n] y(t ) Digital Filter D/A where the digital filter has zero phase delay, i.e. /H( shown below H( ) 1.0 ) 0, and the magnitude response Suppose that the response to the input x(t) = cos 2 (100)t is y(t) = 0.75cos 2 (50)t c c [ ] + cos[ 2 (450)t] [ ] + 0.5cos[ 2 (100) t]. a. (13) Find the unknown sampling frequency f s. b (12) Find the unknown cutoff frequency c for the digital filter.
2 May 2000 2 EE 438 Final Exam 1. (continued)
2 May 2000 3 EE 438 Final Exam 2. (25 pts.) You have just started your new job with SecureCom, Inc. They have developed a speech scrambler that takes an analog speech signal x(t) with CTFT X( f )and swaps the negative and positive sides of the spectrum to produce a signal y(t) with CTFT Y( f ) illustrated by the example below: X( f ) Y( f ) -4kHz 4kHz f -4kHz 4kHz f The shading is included to distinguish between the negative and positive portions of the spectrum. Your assignment is to design a digital system operating at a sampling frequency of 10 khz, which will recover the analog signal x(t) from y(t). The system must operate in real time; so you cannot use a DFT. You can use an A/D converter, D/A converter, digital modulators, digital filters, etc. Your design should clearly specify all the basic parameters of these components, i.e. frequencies, frequency responses, etc.
2 May 2000 4 EE 438 Final Exam 2. (continued)
2 May 2000 5 EE 438 Final Exam 3. (25 pts) You have a radix 2 FFT algorithm that can be used to compute the fast Fourier transform of an N point signal, where N is any power of 2. Suppose that you need to compute a 48 point DFT. a. (15) Derive an expression for the 48 point FFT of a signal that uses your radix 2 FFT. b. (10) Draw a block diagram of your 48 point FFT, showing the radix 2 FFT simply as a box labeled M point radix 2 FFT. Be sure to carefully label all twiddle factors in your diagram.
2 May 2000 6 EE 438 Final Exam 3. (continued)
2 May 2000 7 EE 438 Final Exam 4. (25 pts) The short-time continuous-time Fourier transform (STCTFT) of a speech signal s(t) is defined as S( f,t) = s( )w(t )e j 2 f d, where w(t) is the window function that is used to limit the extent of s( ) to the neighborhood of t. If we plot S( f,t) as intensity vs. time t and frequency f, we obtain a spectrogram. Suppose the signal s(t) consists of a single phoneme with CTFT S( f )given by S( f) 50 f Hz a. (9) Sketch what a narrowband spectrogram would look like. Be sure to dimension all quantities in your sketch. b. (3) What should the duration (in seconds) of the window w(t) be to generate a narrowband spectrogram? (This question does not have a unique answer.) c. (9) Sketch what a wideband spectrogram would look like. Be sure to dimension all quantities in your sketch. d. (3) What should the duration (in seconds) of the window w(t) be to generate a wideband spectrogram? (This question does not have a unique answer.) e. (1) Check here if you would like 1 free point. No one will call you on the phone; and your credit card will not be billed.
2 May 2000 8 EE 438 Final Exam 4. (continued)
2 May 2000 9 EE 438 Final Exam 5. (25 pts.) You have a photograph with an image given by f (x, y) = cos[ 2 (590x +10y) ]. a. (5) Sketch what f (x, y) looks like. You scan this photograph wth your new 600 dots/inch flat bed scanner, and display the image on a high resolution 600 dots/inch monitor. b. (15) Assuming that the scanner acts like an ideal sampler, and the monitor acts like an ideal reconstruction filter with cutoff frequency at 300 dots/inch, find the displayed image g(x, y). c. (5) Sketch what g(x, y) looks like.
2 May 2000 10 EE 438 Final Exam 5. (continued)
2 May 2000 11 EE 438 Final Exam 6. (25 pts.) Consider the image x[m,n] shown below: n 0 0 0 0 0 0 m 0 0 0 0 0 0 0 0 1 1 1 1 0 0 1 1 1 1 0 0 1 1 1 1 0 0 1 1 1 1 Suppose we filter this image with a filter that has impulse response: h[m,n] m n 1 0 1 1 0 1 2 0 0 1 2 1 1 2 1 0 1 2 0 a. (8) Compute the output image y[m,n]. (You may assume that boundary pixels are repeated beyond the boundary.) b. (2) Is the filter DC preserving? c. (8) Find a simple expression for the frequency response H(, ) for this filter. d. (4) Sketch H(, ) for the two cases: i. = 0, ii. = 0, e (3) Discuss the relation between your answers to parts a., b., and d.
2 May 2000 12 EE 438 Final Exam 6. (continued) 1. 2. 3. 4. 5. 6. Total