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20111ee114_1_HW1

# 20111ee114_1_HW1 - x n =[4 1 2(2(a Compute the Z-Transform...

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UCLA Dept. of Electrical Engineering EE 114, Winter 2011 Problem Set 1 Due: January 10, 2011 1. Let x a ( t ) be a continuous-time speech signal 50 ms in duration, that is sampled at F s = 16 kHz. We wish to perform spectral analysis using a radix-2 FFT, with spacing between adjacent frequency bins no greater than 20 Hz. What is the minimum length, N , of the FFT used? If the original signal was 75 ms in duration, would this aﬀect the design of your algorithm? If so, how? 2. Consider the discrete-time signal: x ( n ) = [1 , 2 , 3 , 2 , 1] , (1) where the underscore denotes the origin. Compute the following quantities without explicitly computing the DTFT X ( ω ). Include all steps that led to your answer. (a) Find the phase of X ( ω ). (b) R π - π X ( ω ) (c) R π - π | X ( ω ) | 2 (d) Let y ( n ) = x ( n - n 0 ). Compute Y ( ω ). (e) Let ˆ x ( n ) = x ( n ) e - 0 n . Compute ˆ X ( ω ). 3. Consider the discrete time sequence:
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Unformatted text preview: x ( n ) = [4 , 1 , 2] (2) (a) Compute the Z-Transform X ( z ). (b) Let y ( n ) = x ( n ) * x ( n ). Determine y ( n ) by (i) convolution in the time domain, and (ii) by transformation into the Z-domain. 4. Consider the discrete-time signal: x ( n ) = [1 , , 2 , 0] (3) (a) Compute the 4-point DFT, X ( k ) of the above signal. (b) Using X ( k ), evaluate the inverse transform x ( n ) for n =-6 through 6. 5. Consider a continuous time signal consisting of a cosine at 1 Hertz: x ( t ) = cos (2 πt ) sampled with a sampling frequency of F s = 2 / 3 Hz . The sampled signal is then input to an ideal D/A converter. What is the frequency of the signal that will emerge from the D/A converter? 1...
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