The document discusses sampling theory and its applications. It introduces key concepts such as: 1. Signals can be represented by discrete sample values taken at regular intervals, and reconstructed using an ideal low-pass filter, as described by the sampling theorem. 2. The sampling theorem states that a band-limited signal with no frequencies above B Hz can be uniquely determined by samples taken at least every 1/(2B) seconds. 3. Anti-aliasing filters are used to limit the bandwidth of signals before sampling to avoid aliasing when the sampling rate is lower than predicted by the sampling theorem.

1. 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
greater1 than




 2B 
seconds apart.
EE 541/451 Fall 2006

2. 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(t) A/D fs(t)
conversion
T
Sampling
EE 541/451 Fall 2006

3. Bandpass sampling theory
 2B sampling rate for signal from 2B to 3B
X(f)
-3B -2B -B B 2B 3B
X(f-fs)
-3B -2B -B B 2B 3B 4B 5B
X(f+fs)
-5B -4B -3B -2B -B B 2B 3B
Xs
-5B -4B -3B -2B -B B 2B 3B 4B 5B
EE 541/451 Fall 2006

4. Impulse Sampling
Signal waveform Sampled waveform
0
0
1 201
1 201
Impulse sampler
0
1 201
EE 541/451 Fall 2006

5. Impulse Sampling
with increasing sampling time T
Sampled waveform Sampled waveform
0 0
1 201 1 201
Sampled waveform Sampled waveform
0 0
1 201 1 201
EE 541/451 Fall 2006

6. Introduction
Let gδ (t ) denote the ideal sampled signal
∞
gδ ( t ) = ∑ g (nT ) δ (t − nT )
n = −∞
s s (3.1)
where Ts : sampling period
f s = 1 Ts : sampling rate
EE 541/451 Fall 2006

7. Math
From Table A6.3 we have
∞
g(t ) ∑δ (t − nTs ) ⇔
n =−∞
∞
1 m
G( f ) ∗
Ts
∑ δ( f −
m =−∞ Ts
)
∞
= ∑ f G( f
m =−∞
s − mf s )
∞
gδ ( t ) ⇔ f s ∑G ( f
m =−∞
− mf s ) (3.2)
or we may apply Fourier Transform on (3.1) to obtain
∞
Gδ ( f ) = ∑ g (nT ) exp( − j 2π nf T )
n =−∞
s s (3.3)
∞
or Gδ ( f ) = f sG ( f ) + f s ∑G ( f
m =−∞
− mf s ) (3.5)
m ≠0
If G ( f ) = 0 for f ≥ W and Ts = 1
2W
∞
n jπ n f
Gδ ( f ) = ∑ g ( ) exp( − ) (3.4)
n =−∞ 2W W
EE 541/451 Fall 2006

8. Math, cont.
With
1.G ( f ) = 0 for f ≥W
2. f s = 2W
we find from Equation (3.5) that
1
G( f ) = Gδ ( f ) , − W < f < W (3.6)
2W
Substituting (3.4) into (3.6) we may rewrite G ( f ) as
1 ∞
n jπnf
G( f ) =
2W
∑ 2W
n = −∞
g( ) exp( −
W
) , − W < f < W (3.7)
n
g (t ) is uniquely determined by g ( ) for − ∞ < n < ∞
2W
 n 
or  g ( )  contains all information of g (t )
 2W 
EE 541/451 Fall 2006

9. Interpolation Formula
 n 
To reconstruct g (t ) from  g ( )  , we may have
 2W 
∞
g (t ) = ∫ G ( f ) exp( j 2πft )df
−∞
1 ∞
n jπ n f
∑ g ( 2W ) exp( − W ) exp( j 2π f t )df
W
=∫
−W 2W n = −∞
∞
n 1  n 
= ∑ g(
W
n = −∞
)
2W 2W ∫−W exp  j 2π f (t − 2W )df (3.8)
 
∞
n sin( 2π Wt − nπ )
= ∑ g( )
n = −∞ 2W 2π Wt − nπ
∞
n
= ∑ g( ) sin c( 2Wt − n ) , - ∞ < t < ∞ (3.9)
n = −∞ 2W
(3.9) is an interpolation formula of g (t )
EE 541/451 Fall 2006

10. Interpolation
If the sampling is at exactly the Nyquist rate, then
∞
 t − nTs 
g (t ) = ∑ g (nTs ) sin c
 T  
n = −∞  s 
∞
 t − nTs 
g (t )
g (t ) = ∑ g (nTs ) sin c
 T  
n = −∞  s 
EE 541/451 Fall 2006

11. 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 )
EE 541/451 Fall 2006

12. Sampling Theorem
Sampling Theorem for strictly band - limited signals
1.a signal which is limited to − W < f < W , can be completely
 n 
described by  g ( ) .
 2W 
 n 
2.The signal can be completely recovered from  g ( )
 2W 
Nyquist rate = 2W
Nyquist interval = 1
2W
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.
EE 541/451 Fall 2006

13. Under Sampling, Aliasing
EE 541/451 Fall 2006

14. Avoid Aliasing
 Band-limiting signals (by filtering) before sampling.
 Sampling at a rate that is greater than the Nyquist rate.
Anti-aliasing A/D fs(t)
f(t)
filter conversion
T
Sampling
EE 541/451 Fall 2006

15. Anti-Aliasing
EE 541/451 Fall 2006

16. Aliasing
 2D example
EE 541/451 Fall 2006

17. Example: Aliasing of Sinusoidal Signals
Frequency of signals = 500 Hz, Sampling frequency = 2000Hz
EE 541/451 Fall 2006

18. Example: Aliasing of Sinusoidal Signals
Frequency of signals = 1100 Hz, Sampling frequency = 2000Hz
EE 541/451 Fall 2006

19. Example: Aliasing of Sinusoidal Signals
Frequency of signals = 1500 Hz, Sampling frequency = 2000Hz
EE 541/451 Fall 2006

20. Example: Aliasing of Sinusoidal Signals
Frequency of signals = 1800 Hz, Sampling frequency = 2000Hz
EE 541/451 Fall 2006

21. Example: Aliasing of Sinusoidal Signals
Frequency of signals = 2200 Hz, Sampling frequency = 2000Hz
EE 541/451 Fall 2006

22. Natural sampling
(Sampling with rectangular waveform)
Figure 6.7
Signal waveform Sampled waveform
0
0 1 201 401 601 801 1001 1201 1401 1601 1801 20
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
EE 541/451 Fall 2006

23. Bandpass Sampling
(a) variable sample rate
(b) maximum sample rate without aliasing
(c) minimum sampling rate without aliasing
EE 541/451 Fall 2006

24. Bandpass Sampling
 A signal of bandwidth B, occupying the frequency range
between fL and fL + B, can be uniquely reconstructed from the
samples if sampled at a rate fS :
fS >= 2 * (f2-f1)(1+M/N)
where M=f2/(f2-f1))-N and N = floor(f2/(f2-f1)),
B= f2-f1, f2=NB+MB.
EE 541/451 Fall 2006

25. Bandpass Sampling Theorem
EE 541/451 Fall 2006

26. PAM, PWM, PPM, PCM
EE 541/451 Fall 2006

27. Time Division Multiplexing
 Entire spectrum is allocated for a channel (user) for a limited time.
 The user must not transmit until its
k1 k2 k3 k4 k5 k6
next turn.
 Used in 2nd generation c
Frequency
f
t
 Advantages: Time
– Only one carrier in the medium at any given time
– High throughput even for many users
– Common TX component design, only one power amplifier
– Flexible allocation of resources (multiple time slots).
EE 541/451 Fall 2006

28. Time Division Multiplexing
 Disadvantages
– Synchronization
– Requires terminal to support a much higher data rate than the
user information rate therefore possible problems with
intersymbol-interference.
 Application: GSM
 GSM handsets transmit data at a rate of 270
kbit/s in a 200 kHz channel using GMSK
modulation.
 Each frequency channel is assigned 8 users, each
having a basic data rate of around 13 kbit/s
EE 541/451 Fall 2006