Digital Signal Processing - Sampling and Reconstruction

DIGITAL SIGNAL PROCESSING
Lectured by Assoc. Prof. Dr. Thuong Le-Tien
National distinguished Lecturer
Tel: 0903 787 989, email: thuongle@hcmut.edu.vn
Sept, 2011
Sampling and Reconstruction
Sampling and Reconstruction
• 1. Introduction
• 2. Overview of Analog 
• 3. Sampling theorem
• 4. Sampling of Sinusoids
• 5. Spectra of Sampled 
• 6. Analog signal reconstruction
2
Three steps for digital signal processing of 
analog signals 
 Step 1: Digitizing of analog signals: 
Sampling, Quantization – Analog to Digital 
Conversion (ADC).
 Step 2: Implementing digital signal 
processor for discrete samples
 Step 3: Reconstructing the analog signal 
after processing – Digital to Analog 
Conversion (DAC)
3
1. Introduction
 FOURIER Transform X() of x(t) is the spectrum of the 
analog signal:
(2.1)
 Where  is the radian frequency (rad/s).
and  = 2f (2.2)
 Definition of Laplace Transform:
(2-3)
4
2. Review of Analog signals
dtetxX tj


 )()(
dtetxsX st


 ).()(
 Response of a linear system
 The system is characterized by impulse response h(t). The 
output y(t) is obtained by the time domain convolution :
 Or frequency domain:
where H() is the frequency response of the system. 
5
')'()'()( 


 dttxtthty
)().()(  XHY
Linear system
h(t)
x(t)
input
y(t)
output
 H() is the Fourier transform of h(t)
 The steady state response of a sinusoid:
 Output is a sinusoid with frequency (), 
amplitude equal to the signal amplitude multiplied 
by MagH(), and phase shift equal to arg(H()):
6

 dtethH tj)()(
Linear system
H()
x(t) = exp(jt)
Sinusoid in
y(t) = H()exp(jt)
Sinusoid out
)(arg.|)(|)()()(   Hjtjtjtj eHeHtyetx
 Linear superposition: Signals x(t) has two frequency 
components
 After filtering
 Note: Filtering only change the magnitudes but not 
the frequencies
7
tjtj eAeAtx 21 21)(
 
tjtj eHAeHAty 21 )()()( 21
 
 The result is presented in frequency domain
 Spectrum of X()
 Spectrum of Y()
8
X( )
A 1 A 2
H( )

Y(  )
A 1 H( )

A 2 H( )
)(2)(2)( 2211   AAX
)()(2)()(2
))(2)(2)(()()()(
222111
2211




HAHA
AAHXHY
 Sampling process in Fig. 3.1. x(t) is sampled 
by period T, t=nT where n=0,1,2,
 Many high frequency components appear 
in the signal spectrum 
 Two questions are often provided for 
1. What is the effect of sampling on the 
original frequency spectrum? 
2. How should one choose the sampling 
interval T?
9
3. Concept of Sampling theorem
 The spectrum of the sampled sinusoid x(nT) 
will be periodic replication of the original 
spectral line at intervals fs=1/T 
Figure 3.1 Ideal Sampler
10
11
Figure 3.2. Spectrum replication caused by sampling.
With the replicated spectrum of the sampled signal, one 
cannot tell uniquely What the original frequency was. It 
could be any one of the replicated frequencies namely 
f’=f+mfs.This potential confusion of the original frequency 
with another is known as aliasing and can be avoided if one 
satisfies the condition of the sampling theorem
Sampling theorem
 For accurate representation of a signal x(t) by its 
time samples x(nT), two conditions must be met:
1: x(t) is bandlimited
2: Sampling frequency must be chosen to be 
at least twice the maximum frequency fmax, 
fs  2fmax: 
fs = 2fmax is the Nyquist rate. 
fs/2 is the Nyquist frequency or folding 
frequency
12
Typical sampling rate for some common applications
 Antialiasing Prefilter
 Signal must be bandlimited therefore need to pass 
through a low pass filter namely prefilter before sampling
14
Prefiltered spectrum 
0 
0 - f s f s 
f 
f 
- f s /2 f s /2 
f 
Input spectrum 
prefilter 
Replicated 
spectrum 
Bandlimited 
signal 
x(t) Analog 
signal 
digital 
signal 
x( nT) x(t) Analog lowpass 
filter 
Sampler and 
quantizer 
To DSP 
Antialiasing prefilter
What happens if we do not sample in 
accordance with the sampling theorem?
 Missing important time variations between sampling instants 
 May arrive at the erroneous conclusion that the samples 
represent a signal which is smoother than it actually is
 Be confusing the true frequency content of the signal with a 
lower frequency content. Such confusion of signals is called 
aliasing
Aliasing in 
The time domain
The number of samples per is given by the quantity fs/f:
16
4. Sampling of sinusoid: x(t) = cos(2ft)
cycle
samples
cycles
samples
f
f s 
sec/
sec/
Special case with multiple frequency components in the x(t)
17
Analog reconstruction and aliasing
)()( 222)(2 nTxeeeenTx jfTnTnjmfjfTnTnmffjm
ss   
,...,...,2,, sss mfffffff 
Using the property fsT=1 and the trigonometric identity
Define also the following family of sinusoids, for m in integer
And its sampled version
Note that xm(t) are different from each other 
but they have same samples:
18
LPF as an ideal 
reconstructor
 Example
 As sinusoid f=10 Hz, sampled by fs=12Hz. The sampled 
signal consists of periodic frequencies 10+m.12Hz, m = 0, 
1, 2, or: , -26, -14, -2, 10, 22, 34, 46,  but only fa
= 10 mod(12) = 10 – 12 = -2 Hz in the range of Nyquist
interval [-6,6] Hz. So the reconstructed signal with –2 Hz 
is not as the original one with 10 Hz. 
19
 Example: 5 signals are sampled by the rate 4Hz:
(t second)
Let prove they are aliased each other due to their same 
samples.
 Sol: The frequencies of the signals: -7, -3, 1, 5, 9 Hz. They 
have the same periodic replication in multiples of fs=4Hz.
Writing the five frequencies compactly: 
fm=1+4m, m=-2, -1, 0, 1, 2. 
20
t)sin(18 t),sin(10 t),sin(2 ,)6sin(),t14sin(  t
2-2,-1,0,1,m )),41(2sin()2sin()(  ntftx mm 
)4/2sin()24/2sin( 
)4/)41(2sin())41(2sin()(
nmnn
nmnTmnTxm




Example: x(t)=4+3cos(pt)+2cos(2pt)+cos(3pt) t: in ms
Determine the min sampling rate without any aliasing effects
Supposed the signal sampled at half its Nyquist rate. 
Determine xa(t) that would be aliased with x(t). 
Sol:
Freq. of 4 terms: f1=0, f2=0.5kHz, f3=1kHz,f4=1.5kHz
Example: The square wave sampled at rate fs; t in seconds
Determine the xa(t) that will appear at the output of the 
reconstructor for 2 cases fs=4kHz and 8kHz.
Sol:
Fourier series of square wave contains odd harmonics at freq.
For fs =4kHz, the aliased signal will be
For fs =8kHz, the aliased signal will be
•The first case: Sketch for xa(t)
Condition xa(t)=x(nT) evalued at n=1 implies A=1
•The second case: xa(t)=Bsin(n/4)+Csin(3n/4)
Condition xa(t)=x(nT) at n=1,2 give two equations
Example: A given x(t), t in ms and a block of DSP
Determine the y(t) and ya(t) in the following cases:
a. When there is no prefilter, that is, H(f)=1 for all f
b. When H(f) is the ideal filter with cutoff fs/2=20kHz
c. When H(f) is a practical prefilter as follows,
Sol: Six terms of freq. in x(t)
Case a.
Case b.
Case c.
 Sampled signal: 
 In practical sampling, the sampled signal:
 where, p(t) is flat-top pulse of duration  second. 
Ideal sampling with  toward 0. 
30
5. Spectra of sampled signals




n
nTtnTxtx )()()(ˆ 




n
flat nTtpnTxtx )()()(
0 T 2T . nT t 
0 T 2T . nT t 
x fla t (t) 
 
) ( ˆ t x 
) ( ) ( nT t nT x   
) ( ) ( nT t p nT x  
Discrete Time Fourier Transform DTFT
or
This approximation become exact if 
Practical approximation
32
Spectrum Replication
Aliasing caused by overlapping spectral replicas
Ideal antialiasing prefilter
Practical antialiasing prefilter
Attenuation in dB
35
6. Analog signal reconstruction
Staircase reconstructor
Analog reconstructor as a low pass filter
)(ˆ)()( fYfHfY a 




n
a nTthnTyty )()()(
)(
1
)(ˆ 



m
smffY
T
fY
36




n
nTtnTyty )()()(ˆ 




n
a nTthnTyty )()()(
Replicated spectrum
Reconstructed analog signal
Ideal reconstructor
Staircase reconstructor
Anti-image postfilter
Digital equalization filter for D/A conversion