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Digital Signal and Image Processing using MATLAB, Volume 1: Fundamentals, 2nd Edition by Gerard Blanchet, Maurice Charbit

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Chapter 10

Hints and Solutions

H1 Signal fundamentals

H2 Discrete time signals and sampling

H2.1 (An illustration of the sampling theorem) (see p. 78)

1.Because Fs = 500 Hz is greater than twice the signal’s frequency (that is, 2 × 200 Hz), the sampling makes it possible to perfectly reconstruct the signal. Hence we end up with the same sine at the 200 Hz frequency;
2. because Fs = 250 Hz is smaller than twice the signal’s frequency, the sampling introduces aliasing. The ±Fs shifts in the spectrum (corresponding to n = ±1 in formula 2.6) contribute to the frequency with –250 + 200 = 50 Hz. Since the spectrum is symmetrical, everything happens as if the 200 Hz frequency were “aliased” by symmetry about the frequency Fs/2 = 125 Hz. The result of the reconstruction is a sine with the frequency 50 Hz (Figure H2.1);
3. type:
images
images

Figure H2.1Sampling and reconstruction

images

H2.2 (Time domain hermitian symmetry) (see p. 86)

1. If we take the conjugate complex of X(f) and use x(n) = x*(–n):
images
If, furthermore, x(n)is real, then we know that X(f ) = X* (–f ), hence X(f ) = X(–f ) = X* (f ). The conclusion ...

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