6. Sampling Theorem
(For Lowpass Signals)
Let the finite energy signal g(t) is bandlimited to W; i.e., G(f) ≡ 0
for │f│≥ W. Let g(t) be sampled at multiples of some basic
sampling interval Ts , where Ts ≤ 1/2W, to yield the sequence
{g(nTs)} for -∞ ≤ n ≤ ∞. Then it is possible to reconstruct the
original signal g(t) from the sampled values without any
distortion.
Sampling rate fs = 2W is called Nyquist rate
13. Spectrum of analog signal and under-
sampled signal showing aliasing
fmax
Frequency
Amplitude
aliasing
Frequency
Amplitude
Spectral Window
fs/2
fs
(a)
(b)
14.
15. (a) Anti-alias filtered spectrum of signal
(b) Spectrum of instantaneously sampled signal, assuming sampling rate greater
than Nyquist rate
(c) Magnitude response of reconstruction filter
16. (a) Anti-alias filtered spectrum of signal
(b) Spectrum of instantaneously sampled signal, assuming sampling
rate greater than Nyquist rate
(c) Magnitude response of reconstruction filter
17. Sampling Theorem For Bandpass Signals
• Consider a bandpass signal occupying a
frequency band fL to fH. The minimum sampling
frequency fs allowable is 2(fH – fL) provided that
either fH or fL is an integer multiple of fs.
18. Sampling Theorem For Random Process
Consider a WSS random process, X(t),
bandlimited to W Hz {i.e., SX(f)=0 for │f│≥W}.
To reconstruct the process from its samples with
zero mean square error, {E│X(t) – Xꞌ(t)│2}→0,
the minimum sampling rate required will be 2W.
19. Analog Pulse Modulation
• Pulse amplitude modulation (PAM)
• Pulse width modulation (PWM)
• Pulse position modulation (PPM)
32. Sample and Hold Circuit For Signal
Recovery
• The reconstruction can be done by
interpolating between samples.
• Ideal low-pass filter can do interpolation.
• Practical systems use S/H circuits
– Zero-order-hold (ZOH)
– First-order-hold (FOH)
