Transcript ppt
ECE 4371, Fall, 2014 Introduction to Telecommunication Engineering/Telecommunication Laboratory
Zhu Han Department of Electrical and Computer Engineering Class 7 Sep. 17 th , 2014
Outline
Analog vs. Digital ADC/DAC: gateway between analog and digital domains – Sampling Theorem – Quantization – Most important part in communication system – Most important during interview – Read books carefully Examples
Claude Elwood Shannon, Harry Nyquist
Sampling Theory
In many applications it is useful to represent a signal in terms of sample values taken at appropriately spaced intervals.
The signal can be reconstructed from the sampled waveform by passing it through an ideal low pass filter.
In order to ensure a faithful reconstruction, the original signal must be sampled at an appropriate rate as described in the
sampling theorem
. – A real-valued band-limited signal having no spectral components above a frequency of B Hz is determined uniquely by its values at uniform intervals spaced no greater than seconds apart.
2B
Sampling Block Diagram
Consider a band-limited signal f(t) having no spectral component above B Hz.
Let each rectangular sampling pulse have unit amplitudes, seconds in width and occurring at interval of T seconds.
f s
(
t) f
(
t
)
A/D conversion
T
Sampling
0 1
Impulse Sampling
Signal waveform Sampled waveform
0 1 201
Impulse sampler
201 0 1 201
Impulse Sampling with increasing sampling time T
Sampled waveform Sampled waveform
0 1
Sampled waveform
201 0 1
Sampled waveform
201 0 1 201 0 1 201
Introduction
Let
g
(
t
) denote the ideal sampled signal
g
(
t
)
n
g
(
nT s
) (
t
nT s
) (3.1) Equation number is not the same as in the book where
T s f s
: sampling 1
T s
: period sampling rate
Math
From Table A6.3
we have g(
t
)
n
(
t G
(
f
) 1
T s m
f s G
(
m
(
f f
mf s
)
g
(
t
)
nT s
)
f s m
G
(
f m
)
T s
mf s
) (3.2) or we may apply Four ier Transform on (3.1) to obtain
G
( or
G
(
f f
) )
n
g
(
nT s
) exp(
f s G
(
f
)
j
2
nf T s
) (3.3)
f s
m m
0
G
(
f
mf s
) (3.5) If
G
(
G
(
f f
) ) 0 for
n
g
(
f
n
2
W W
and ) exp(
T s
1 2
W j
n f
) (3.4)
W
Math, cont.
With 1 .
G
(
f
) 0 for 2 .
f s
2
W f
W
we find from Equation (3.5) that
G
(
f
) 1 2
W G
(
f
) ,
W
f
W
(3.6) Substituti ng (3.4) into (3.6) we may rewrite
G
(
f
) as
G
(
f
) 1 2
W n
n g
( 2
W
) exp(
j
nf
) ,
W
f
W
(3.7)
W g
(
t
) is uniquely determined
n
by
g
( 2
W
) for
n
n
or
g
( 2
W
) contains all informatio n of
g
(
t
)
Interpolation Formula
To reconstruc t
g
(
t
) from
g n
( 2
W
)
g
(
t
)
G
(
f
) exp(
j
2
ft
)
df
W
W
1 2
W n
n g
( 2
W
) exp(
W
, we may have
j
n f
) exp(
j
2
f t
)
df
n
n g
( 2
W
n
n g
( 2
W
n
n g
( 2
W
) ) 1 2
W
W
W
) sin
c
( 2
Wt
exp
j
sin( 2
Wt
2
Wt
n n
2
f
) (
t
n
2
W
)
df
(3.8)
n
) ,
t
(3.9) (3.9) is an interpolat ion formula of
g
(
t
)
Interpolation
If the sampling is at exactly the Nyquist rate, then
g
(
t
)
n
g
(
nT s
) sin
c
t T s nT s
g
(
t
)
g
(
t
)
n
g
(
nT s
) sin
c
t T nT s
s
Practical Interpolation
Sinc-function interpolation is theoretically perfect but it can never be done in practice because it requires samples from the signal for all time. Therefore real interpolation must make some compromises. Probably the simplest realizable interpolation technique is what a DAC does.
g
(
t
)
Sampling Theorem
Sampling 1.a signal Theorem which is for strictly limited to band
W
f
limited
W
signals , can be completely described by
n g
( 2
W
) .
2 .
The signal can be completely recovered
n
from
g
( 2
W
) Nyquist rate 2
W
Nyquist interval 1 2
W
When the signal is not band limited (under sampling) aliasing occurs .To
avoid aliasing, we may limit the signal bandwidth or have higher sampling rate.
Under Sampling, Aliasing
Avoid Aliasing
Band-limiting signals (by filtering) before sampling.
Sampling at a rate that is greater than the Nyquist rate.
f
(
t
)
Anti-aliasing filter A/D conversion
f s (t) T
Sampling
Anti-Aliasing
2D example
Aliasing
Example: Aliasing of Sinusoidal Signals
Frequency of signals = 500 Hz, Sampling frequency = 2000Hz
Example: Aliasing of Sinusoidal Signals
Frequency of signals = 1100 Hz, Sampling frequency = 2000Hz
Example: Aliasing of Sinusoidal Signals
Frequency of signals = 1500 Hz, Sampling frequency = 2000Hz
Example: Aliasing of Sinusoidal Signals
Frequency of signals = 1800 Hz, Sampling frequency = 2000Hz
Example: Aliasing of Sinusoidal Signals
Frequency of signals = 2200 Hz, Sampling frequency = 2000Hz
Natural sampling (Sampling with rectangular waveform)
Figure 6.10
Signal waveform Sampled waveform
0 1 201 401 601 801 1001 1201 1401 1601 1801 2001
Natural sampler
0 1 201 401 601 801 1001 1201 1401 1601 1801 2001 0 1 201 401 601 801 1001 1201 1401 1601 1801 2001
Bandpass Sampling
(a) variable sample rate (b) maximum sample rate without aliasing (c) minimum sampling rate without aliasing
Bandpass Sampling
A signal of bandwidth B, occupying the frequency range between f L and f L + B, can be uniquely reconstructed from the samples if sampled at a rate f S : f S >= 2 * (f2-f1)(1+M/N) where M=f 2 /(f 2 -f 1 ))-N and N = floor(f 2 /(f 2 -f 1 )), B= f 2 -f 1, f2=NB+MB.
Bandpass Sampling Theorem