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| + | #'''Sampling''' | ||
| + | ##Impulse Train Sampling | ||
| + | ##The Sampling Theorem and the Nyquist | ||
| + | #'''Signal Reconstruction Using Interpolation:''' the fitting of a continuous signal to a set of sample values | ||
| + | ##Sampling with a Zero-Order Hold (Horizontal Plateaus) | ||
| + | ##Linear Interpolation (Connect the Samples) | ||
| + | #Undersampling: Aliasing | ||
| + | #Processing CT Signals Using DT Systems (Vinyl to CD) | ||
| + | ##Analog vs. Digital: The Show-down (A to D conversion -> Discrete-Time Processing System -> D to A conversion | ||
| + | #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 <math>X(j\omega) = 0</math> for <math>|\omega| > \omega_M</math>. Then x(t) is uniquely determined by its samples <math>x(nT), n = 0, \pm 1, \pm 2,..., \mbox{ if}</math> | ||
| + | : | ||
| + | :::<math>\,\!\omega_s > 2\omega_M</math> | ||
| + | :where | ||
| + | :::<math>\omega_s = \frac{2\pi}{T}</math> . | ||
| + | |||
| + | :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 <math>X(\omega)</math> the Fourier transform of x(t). | ||
| + | : | ||
| + | |||
| + | |||
| + | :<math>\,\!p(t) = \sum_{n = -\infty}^{+\infty}\delta (t - nT)</math> | ||
| + | :: | | ||
| + | :: | | ||
| + | :: 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 | ||
Latest revision as of 06:05, 8 December 2008
Chapter 7
- Sampling
- Impulse Train Sampling
- The Sampling Theorem and the Nyquist
- Signal Reconstruction Using Interpolation: the fitting of a continuous signal to a set of sample values
- Sampling with a Zero-Order Hold (Horizontal Plateaus)
- Linear Interpolation (Connect the Samples)
- Undersampling: Aliasing
- Processing CT Signals Using DT Systems (Vinyl to CD)
- Analog vs. Digital: The Show-down (A to D conversion -> Discrete-Time Processing System -> D to A conversion
- 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) $
- |
- |
- 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
