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Chapter 7

1. Sampling
   1. Impulse Train Sampling
   2. The Sampling Theorem and the Nyquist
2. Signal Reconstruction Using Interpolation: the fitting of a continuous signal to a set of sample values
   1. Sampling with a Zero-Order Hold (Horizontal Plateaus)
   2. Linear Interpolation (Connect the Samples)
3. Undersampling: Aliasing
4. Processing CT Signals Using DT Systems (Vinyl to CD)
   1. Analog vs. Digital: The Show-down (A to D conversion -> Discrete-Time Processing System -> D to A conversion
5. Sampling DT Signals (CD to MP3 albeit a complicated sampling algorithm, MP3 is less dense signal)

Sampling Theory

Let x(t) be a band-limited signal with $ X(j\omega) = 0 $ for $ |\omega| > \omega_M $. Then x(t) is uniquely determined by its samples $ x(nT), n = 0, \pm 1, \pm 2,..., \mbox{ if} $

$ \,\!\omega_s > 2\omega_M $

where

$ \omega_s = \frac{2\pi}{T} $ .

Given these samples, we can reconstruct x(t) by generating a periodic impulse train in which successive impulses have amplitudes that are successive sample values. This impulse train is then processed through an ideal lowpass filter with gain T and cutoff frequency greater than \omega_M and les than \omega_s - \omega_M. The resulting output signal will exactly equal x(t)

Observations

To determine if x(t) is band-limited, one must exam $ X(\omega) $ the Fourier transform of x(t).

$ \,\!p(t) = \sum_{n = -\infty}^{+\infty}\delta (t - nT) $

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v

- - ->X- - ->|H(j\omega)|- - -> x_r(t)

Recommended Exercises: 7.1, 7.2, 7.3, 7.4, 7.5, 7.7, 7.10, 7.22, 7.29, 7.31, 7.33