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Project+1+Fall+2011-1 - Project 1 Sampling(v1.1 Common...

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Project 1: Sampling (v1.1) Common Sense Caveat: If you need to modify a problem in order to obtain a meaningful outcome, do so but do so using common sense. The object of the Project 1 is to provide an inquiry-based experience in exploring sampling and sampling limitations. The study makes direct use of Shannon’s Sampling Theorem but requires that you properly interpret this important signal processing tool. Part 1.a: Implement, observe, quantify, and qualify the composite sinusoidal signal. Construct the audio signal x[k] defined below, sampled at f s = 44,100Sa/s, with a signal duration around 5 seconds (choose a signal length N to be a radix-2 number). The signal x[k] consists of 5 distinct sinusoidal components satisfying: ( 29 ( 29 ( 29 ( 29 ) ( ] 14700 , 441 [ ; ) / 2 cos ] [ 689 64 / ; ) / 2 cos ] [ amplitude decreasing ; ] [ 1 / 1 ] [ ] [ 0 0 0 4 1 0 random uniformly Hz f f k f k x Hz f f f k f k x k x i k x k x i s i i s s i i = = = = + + = = π π Analyze each individual signal spectrum and listen to x[k] (MATLAB SOUND). MATLAB Code Snippet » nn=0:1:1023; » fs=44100; f0=fs/64; % 689 Hz » x=cos(2*pi*f0*nn/fs); % cosine » [X,w]=freqz(x); % generate spectrum - could also have used X=abs(fft(x)) » plot(nn,x); plot(w,abs(X)); Figure 1: 1024-point cosine sampled at f s = 44100 Sa/s centered at ϖ 0 = 2 π f 0 where f 0 = 689 Hz (f s /64). The normalized Nyquist frequency is π .
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