Sampling of
Continuous-Time Signals
Dr. Ray Kwok
SJSU
Fall 2013
Continuous-to-Discrete-time
x[n]
sequence of samples
xc(t)
continuous-time signal
x[n] = xc(nT)
sequence
T
sampling period
fs = 1/T
sampling frequency
Dr. Ray Kwok
Sampling of Continuous-Time Signals
In time domain…
fs >> fo, easy recovery
fo
t
Dr. Ray Kwok
fs = 2fo
Ok…. Can recover
fo
t
Dr. Ray Kwok
fs<2fo
Dr. Ray Kwok
Might not recover the original signal
Dr. Ray Kwok
Nyquist Sampling Theorem
Let xc(t) be a bandlimited signal, i.e. Xc(jΩ) = 0 for |Ω| ≥ ΩN.
Then xc(t) is uniquely determined by its samples x[n] = xc(nT) if
ΩN is called the Nyquist frequency.
2ΩN is referred to as the Nyquist rate.
Dr. Ray Kwok
Ωs ≡
2π
≥ 2Ω N .
T
Sampling Rate
∞
s(t ) =
∑ δ (t − nT )
impulse train
n = −∞
∞
xs (t ) ≡ xc (t ) s(t ) =
∑ x (nT )δ (t − nT )
c
n = −∞
Less samples
xs(t) still has the time info.
x[n] only a sequence of n.
Otherwise, they are the same.
shrink
Figure 4.2 Sampling with a periodic impulse train,
followed by conversion to a discrete-time sequence.
(a) Overall system. (b) xs (t) for two sampling rates.
(c) The output sequence for the two different
sampling rates.
Dr. Ray Kwok
Sampling of Continuous-Time Signals
Frequency-Domain of sampling
∞
s(t ) =
∑ δ (t − nT )
impulse train
original
signal
n = −∞
2π
S ( jΩ ) =
T
∞
∑ δ (Ω − kΩ )
s
k = −∞
∞
xs (t ) ≡ xc (t ) s(t ) =
∑ x (nT )δ (t − nT )
c
n = −∞
1
X c ( jΩ ) ∗ S ( j Ω )
2π
1 ∞
X s ( jΩ) = ∑ X c ( j (Ω − kΩ s ))
T k = −∞
X s ( jΩ ) =
Figure 4.3 Frequency-domain representation of sampling in the time
domain. (a) Spectrum of the original signal. (b) Fourier transform of the
sampling function. (c) Fourier transform of the sampled signal with Ωs
> 2ΩN. (d) Fourier transform of the sampled signal with Ωs < 2ΩN.
Dr. Ray Kwok
Sampling of Continuous-Time Signals
Ω s > 2Ω N
Ω s < 2Ω N
Recovery
LPF
original
signal
Ideal low-pass filter
Figure 4.4 Exact recovery of a continuoustime signal from its samples using an ideal
lowpass filter.
Dr. Ray Kwok
Sampling of Continuous-Time Signals
recovered
signal
Aliasing
example
xc (t ) = cos Ω ot
xr (t ) = cos Ω ot
Ω s > 2Ω o
xr (t ) = cos[(Ω s − Ω o )t ]
Figure 4.5 The effect of aliasing in
the sampling of a cosine signal.
Ωs
< Ωo < Ω s
2
Dr. Ray Kwok
Sampling of Continuous-Time Signals
Ideal Filter
not so ideal….
Ideal “low-pass” filter
sinc
Figure 4.7 (a) Block diagram of an ideal band limited
signal reconstruction system. (b) Frequency response
of an
ideal reconstruction filter. (c) Impulse response of an
ideal reconstruction filter.
Dr. Ray Kwok
Sampling of Continuous-Time Signals
Typical signal
Sum of all filtered response
Figure 4.8 Ideal band limited interpolation.
Dr. Ray Kwok
Sampling of Continuous-Time Signals
Next…..
Filtering (digital & analog)
Dr. Ray Kwok