Download Signal Processing

• Name: Dr. Peter Tsang
• Room: G6505
• Ext: 7763
• Email: [email protected]
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Introduction to Digital Signal Processing
Digital Filter Design
Multi-rate Signal Processing
Wavelet
Applications
http://www.ee.cityu.edu.hk/~csl/adsp/adsp.html
• A.N. Akansu et. al., “Multiresolution Signal Decomposition”, Academic
Press.
• P.P. Vaidyanathan, “Multirate Systems and Filter Banks”, Prentice Hall.
Students are strongly recommended to look for reference books in the
library
Special Arrangement
Class cancellation: Week 5 (6th Oct 2007)
Course Assessment: 100%
Laboratory Sessions :
1st weeks 8
2nd weeks 10 to 13
Subject to change if necessary
Expected outcome from students:
1.
2.
3.
4.
5.
Familiarize with FIR and IIR Digital Filter Design.
Establish the concept of Multi-resolution Signal Decomposition.
Understanding the basic mathematical framework of Wavclet
Decomposition.
Capable of designing and building Digital Filters, Multi-resolution and
Wavelet filter banks.
Applying Wavelet Decomposition in image processing.
Signal Processing
x(t)
Input
signal
System
y(t) =
G{x(t)}
Output
signal
Transfer function (filter)
Allow certain frequency band
to pass, and reject others
Figure 1
Signal Processing
x(t)
hFF(t)
y(t) =
G{x(t)}
Output
signal
Input
signal
Non-recursive system
hFF(t)
Feed forward response
Figure 2
Signal Processing
x(t)
y(t) =
G{x(t)}
hFF(t)
Output
signal
Input
signal
Non-recursive system
y(t) = x(t) * hFF(t)
(1)
Figure 3

Convolution
y(t) = x(t) * h (t)  y(t )   h  x t    d

h()
x( )

-2 -1 0 1 2 3 4 5 6

-2 -1 0 1 2 3 4 5 6
y(0)= AREA[h()
 x(-)]

-5 -4 -3 -2 -1 0 1 2 3 4 5 6
y(1)= AREA[h()

-5 -4 -3 -2 -1 0 1 2 3 4 5 6
 x(1-)]

Convolution
y(t) = x(t) * h (t)  y(t )   h  x t    d

h()
x( )

-2 -1 0 1 2 3 4 5 6

-2 -1 0 1 2 3 4 5 6
y(2)= AREA[h()
 x(2-)]

-5 -4 -3 -2 -1 0 1 2 3 4 5 6
y(3)= AREA[h()

-5 -4 -3 -2 -1 0 1 2 3 4 5 6
 x(3-)]
Signal Processing
hFF(t)
+
x(t)
Input
signal
hFB(t)
y(t) =
G{x(t)}
Output
signal
Recursive system
hFB(t)
Feed backward response
Figure 4
Signal Processing
hFF(t)
+
y(t) =
G{x(t)}
x(t)
Input
signal
hFB(t)
Output
signal
Recursive system
y(t) = x(t) * hFF(t) + y(t) * hFB(t)
(2)
Figure 5
Signal Processing
Analog systems are implemented with analog
circuits built up with resistors, capacitors,
inductors and transistors.
Both input and output signals are continuous
waveforms.
Digital Signal Processing
- a five steps process
Step 1 - Sampling
TS
x(t)
x(nTS)
Figure 6
Digital Signal Processing
- a five steps process
Step 2 - Analog to Digital Conversion (ADC)
x(nTS)
x(n) = [x(0) , x(1) , x(2) , ............ , x(N-2) , x(N-1)]
Figure 7
Digital Signal Processing
- a five steps process
Step 3 - Computation on input data
sequence
e.g. A simple Low Pass Filter
y(n) = 0.25x(n-1) + 0.5x(n) + 0.25x(n+1)
(3)
Digital Signal Processing
- a five steps process
Step 4 - Digital to Analog Conversion (DAC)
y(n) = [y(0) , y(1) , y(2) , ............ , y(N-2) , y(N-2)]
y(nTS)
Figure 8
Digital Signal Processing
- a five steps process
Step 5 Interpolation
y(nTS)
y(t)
Figure 9
Digital Signal Processing
x(n)
x(t)
Digital
Filter
y(n)
ADC
y(t)
DAC
+ Int.
y(n) = G{x(n)}
Figure 10
Digital Signal Processing
- key issues
1. Build a mathematical model of the system.
2. Design algorithms and formulations for the
model
3. Apply the algorithms to the input data and
calculate the output data
4. convert the data to the time domain
Models and Algorithms
1. A model is a mathematical description on the
response of the system
2. An algorithm is the realization of the model
Model: H(s) = 1/(s+b)
R
Analog realization
C
Models and Algorithms
1. A model is a mathematical description on the
response of the system
2. An algorithm is the realization of the model
Model: H(s) = 1/(s+b)
Digital realization
y(n) = w1x(n-1) + w2x(n) + w3x(n+1)
Models and Algorithms
Analog System: Determine transfer function H(s)
Built circuit equivalent to H(s)
Digital system: Given H(s), how to determine the
equation and parameters?
Solution: A standardized equation for the class of
linear time-invariant (LTI) systems.
What is an LTI system?
Linear
x1(n)
x2(n)
Digital
Filter
Digital
Filter
y1(n)
y2(n)
Figure 11
What is an LTI system?
Linear
x1(n)
x2(n)
x1(n) + x2(n)
Digital
Filter
Digital
Filter
Digital
Filter
y1(n)
y2(n)
y1(n) + y2(n)
Figure 12
What is an LTI system?
Time Same response to every part of the
Invariant input sequence
x (n)
Digital
Filter
y (n) = G{x(n)}
x (n-no)
Digital
Filter
y (n-no) = G{x(n-no)}
Figure 13
Generalised LTI system
M
NF
y (n) =  ak y(n-k) +  bk x(n-k)
k=1
k= -NF
(4)
Generalised LTI system
M
NF
y (n) =  ak y(n-k) +  bk x(n-k)
k=1
(4)
k= -NF
Modifying parameters { M, NF, ak and bk}
gives different responses (filtering functions).
Summary
Designing analog systems
• Identify the desire filter response (e.g. HP, LP, etc.)
Summary
Designing analog systems
• Identify the desire filter response (e.g. HP, LP, etc.)
• Determine the mathematical representation of the
response H(s)
Summary
Designing analog systems
• Identify the desire filter response (e.g. HP, LP, etc.)
• Determine the mathematical representation of the
response H(s)
• Implement the filter circuit with RLC transistors
and FETs
Summary
Designing digital systems
• Digitize the input analog signal into a sequence of
data x(n) = [x(0), x(1), x(2), ........x(N-1)]
Summary
Designing digital systems
• Digitize the input analog signal into a sequence of
data x(n) = [x(0), x(1), x(2), ........x(N-1)]
• Identify the desire digital filter response (e.g. HP,
LP, etc.)
Summary
Designing digital systems
• Digitize the input analog signal into a sequence of
data x(n) = [x(0), x(1), x(2), ........x(N-1)]
• Identify the desire digital filter response (e.g. HP,
LP, etc.)
• Determine the mathematical representation of the
digital response G(n)
Summary
Designing digital systems
• Digitize the input analog signal into a sequence of
data x(n) = [x(0), x(1), x(2), ........x(N-1)]
• Identify the desire digital filter response (e.g. HP,
LP, etc.)
• Determine the mathematical representation of the
digital response G(n)
• Implement the digital filter with the Generalised
LTI architecture
It sounds simple,
but something does not fit in
What is it?
It sounds simple,
but something does not fit in
What is it?
The sequence of data
x(n) = [x(0), x(1), x(2), ........x(N-1)]
seems to be unrelated to time and frequency.
In fact, the data may have nothing to do with time.
e.g., the intensity of a row of pixels in an image
x(0) = 255
x(1) = 255
x(2) = 128
x(3) = 128
x(4) = 255
x(5) = 255
x(n)= [ 255,255,128,128,255,255]
Figure 14
So what is meant by frequency in the
digital domain?
So what is meant by frequency in the
digital domain?
Number of cycles or repetitions within a
sequence of data
Number of cycles or repetitions within a
sequence of data.
Consider a 12 points sampling lattice
x(n)
3
2
1
0
-1
-2
-3
1 cycle
n
Figure 15
Number of cycles or repetitions within a
sequence of data.
Consider a 12 points sampling lattice
x(n)
3
2
1
0
-1
-2
-3
2 cycle
n
Figure 16
Maximum number of cycles that can be
represented in a sampling lattice.
Consider a 12 points sampling lattice
x(n)
3
2
1
0
-1
-2
-3
n
Figure 17
Number of cycles or repetitions within a
sequence of data.
Maximum number of cycles = N/2
x(n)
3
2
1
0
-1
-2
-3
n
Figure 18
Assuming that the sampling frequency
is 1 Hertz or 2radians/second
The maximum frequency that can be
represented is 1/2 Hertz or
radians/second
Assuming that the sampling frequency
is 1 Hertz or 2radians/second
The maximum frequency that can be
represented is 1/2 Hertz or
radians/second
The resolution in the frequency domain
is 1 cycle, i.e. 1/N Hertz or 2
A typical spectrum
Mirror
Image
x(n)
Mirror
Image
Mirror
Image
f
0
2
N

2
N

2
N

2
N
Figure 19
•
The bandwidth of any set of sequence is
restricted to [0,  ]
•
•
The bandwidth of any set of sequence is
restricted to [0,  ]
The resolution in the frequency domain is
2 /N
•
•
•
The bandwidth of any set of sequence is
restricted to [0,  ]
The resolution in the frequency domain is
2 /N
All real sequences have symmetrical USB and
LSB
•
•
•
•
The bandwidth of any set of sequence is
restricted to [0,  ]
The resolution in the frequency domain is
2 /N
All real sequences have symmetrical USB and
LSB
The ‘frequency’ of the sequence is relative to
the sampling frequency which is taken as 1 Hz
or 2 radians/second
•
•
•
•
The bandwidth of any set of sequence is
restricted to [0,  ]
The resolution in the frequency domain is
2 /N
All real sequences have symmetrical USB and
LSB
The ‘frequency’ of the sequence is relative to
the sampling frequency which is taken as 1 Hz
or 2 radians/second
How to obtain the Spectrum from a
Sequence of Data?
•
For analog waveform, we use Fourier
Transform

Forward
Transform
X ( j ) 
Inverse
Transform
1
x (t ) 
2
 x (t )e
 jt
dt
(5)
d
(6)


 X ( j )e

jt
How to obtain the Spectrum from a
Sequence of Data?
•
For digital sequence, we use Discrete Fourier
Transform
Forward
Transform
N 1
N 1
n 0
n 0
X (k )   x(n)e j 2nk / N   x(n)e  jk n (7)
N 1
N 1
Inverse
Transform
x(n)   X (k )e j 2nk / N   X (k )e jk n
(8)
with
 k  2k / N
(9)
k 0
k 0
What are ‘n’ and ‘k’?
Forward
Transform
N 1
N 1
n 0
n 0
X (k )   x(n)e j 2nk / N   x(n)e  jk n
n is an index to each sample of the waveform xn 
3
2
1
0
-1
-2
-3
n
n=1 n=2 n=3
What are ‘n’ and ‘k’?
Forward
Transform
N 1
N 1
n 0
n 0
X (k )   x(n)e j 2nk / N   x(n)e  jk n

k is an index to each frequency component in theX spectrum
k
Each frequency component is calculated with a value of ‘k’, e.g.
The first (D.C.) component corresponds to k=0
X (0 ) 
N 1
 x( n)e
 j 2n 0 / N
n 0

N 1
 x( n)
n 0
The second component corresponds to k=1, and so on
X (1 ) 
N 1
 x(n)e
n 0
 j 2n1 / N

N 1
 j 2n / N
x
(
n
)
e

n 0
How to obtain the Spectrum from a
Sequence of Data?
•
For digital filter design, the z Transform is
often used as well. (See suplementary notes)
Forward z
Transform
X ( z) 

 x( n ) z
n  
n
(10)
Unlike Fourier Transform, there is no simple inverse transform
relation for z Transform.
How to obtain the Spectrum from a
Sequence of Data?
•
For digital filter design, the z Transform is
often used as well
Forward z
Transform
X ( z) 

 x( n ) z
n  
n
(10)
Unlike Fourier Transform, there is no simple inverse
transform relation for z Transform.
z is a complex variable which can be represented as:
z = rej
The basic meaning of ‘z’
Img
z = rej
r

Real
A 2-D variable
Figure 20