Name: Dr. Peter Tsang Room: G6505 Ext: 7763 Email: [email protected]

• Name: Dr. Peter Tsang

• Room: G6505

• Ext: 7763

• Email: [email protected]

• 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

Course Assessment: 100%Laboratory Sessions : 1st weeks 82nd weeks 10 to 13

Subject to change if necessary

Special ArrangementClass cancellation: Week 5 (6th Oct 2007)

Expected outcome from students:

1. Familiarize with FIR and IIR Digital Filter Design.2. Establish the concept of Multi-resolution Signal Decomposition.3. Understanding the basic mathematical framework of Wavclet

Decomposition.4. Capable of designing and building Digital Filters, Multi-resolution and

Wavelet filter banks.5. Applying Wavelet Decomposition in image processing.

x(t) Systemy(t) = G{x(t)}

Input signal

Transfer function (filter)Allow certain frequency band

to pass, and reject others

Outputsignal

Figure 1

Signal ProcessingSignal ProcessingSignal ProcessingSignal Processing

x(t)

Input signal

Non-recursive system

Outputsignal

hFF(t)

hFF(t) Feed forward response

y(t) = G{x(t)}

Figure 2


x(t)

Input signal

Non-recursive system

Outputsignal

hFF(t)

y(t) = x(t) * hFF(t) (1)

y(t) = G{x(t)}

Figure 3


ConvolutionConvolutionConvolutionConvolution y(t) = x(t) * h (t)

0 1 2 3 4 5 6-1-2

h()

0 1 2 3 4 5 6-1-2

x( )

0 1 2 3 4 5 6-1-2

y(0)= AREA[h() x(-)]

-3-4-5

y t h x t d( )

0 1 2 3 4 5 6-1-2

y(1)= AREA[h() x(1-)]

-3-4-5

ConvolutionConvolutionConvolutionConvolution y(t) = x(t) * h (t)

0 1 2 3 4 5 6-1-2

h()

0 1 2 3 4 5 6-1-2

x( )

0 1 2 3 4 5 6-1-2

y(2)= AREA[h() x(2-)]

-3-4-5

y t h x t d( )

0 1 2 3 4 5 6-1-2

y(3)= AREA[h() x(3-)]

-3-4-5

x(t)

Input signal

Recursive system

Outputsignal

hFF(t)

hFB(t)

+

hFB(t) Feed backward response

y(t) = G{x(t)}

Figure 4


y(t) = x(t) * hFF(t) +

x(t)

Input signal

Recursive system

Outputsignal

hFF(t)

hFB(t)

+

y(t) * hFB(t) (2) Figure 5

y(t) = G{x(t)}


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 Digital Signal Processing - a five steps process- a five steps process


Step 1 - Sampling

x(t)

x(nTS)

TS

Figure 6

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



Step 3 - Computation on input data sequence

y(n) = 0.25x(n-1) + 0.5x(n) + 0.25x(n+1) (3)

e.g. A simple Low Pass Filter



Step 4 - Digital to Analog Conversion (DAC)

y(nTS)

y(n) = [y(0) , y(1) , y(2) , ............ , y(N-2) , y(N-2)]

Figure 8



Step 5 Interpolation

y(t)

y(nTS)

Figure 9



x(t)x(n)

ADC

Digital Filter

DAC + Int.

y(n)y(t)

y(n) = G{x(n)}

Figure 10

Digital Signal ProcessingDigital Signal ProcessingDigital Signal ProcessingDigital Signal Processing

Digital Signal Processing Digital Signal Processing - key issues- key issues

Digital Signal Processing Digital Signal Processing - key issues- 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 AlgorithmsModels and AlgorithmsModels and AlgorithmsModels 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

CAnalog realization

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)


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?What is an LTI system?What is an LTI system?What is an LTI system?

x1(n) Digital Filter

Linear y1(n)


y2(n)

Figure 11


Linear y1(n)


y2(n)

x1(n) + x2(n) Digital Filter

y1(n) + y2(n)

Figure 12


Time Invariant

Same response to every part of the input sequence

Digital Filter

y (n) = G{x(n)}

x (n-no) Digital Filter

y (n-no) = G{x(n-no)}

x (n)

Figure 13


Generalised LTI systemGeneralised LTI systemGeneralised LTI systemGeneralised LTI system

(4)y (n) = ak y(n-k) +

k=1

Mbk x(n-k)

k= -NF

NF

y (n) = ak y(n-k) +

k=1

Mbk x(n-k)

k= -NF

NF

Modifying parameters { M, NF, ak and bk}gives different responses (filtering functions).

(4)

Generalised LTI systemGeneralised LTI systemGeneralised LTI systemGeneralised LTI system

SummarySummaryDesigning analog systemsDesigning analog systems


• Identify the desire filter response (e.g. HP, LP, etc.)

• Identify the desire filter response (e.g. HP, LP, etc.)• Determine the mathematical representation of the

response H(s)



• 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



• Digitize the input analog signal into a sequence of data x(n) = [x(0), x(1), x(2), ........x(N-1)]

SummarySummaryDesigning digital systemsDesigning digital systems



• Identify the desire digital filter response (e.g. HP, LP, etc.)





• Determine the mathematical representation of the digital response G(n)





• Determine the mathematical representation of the digital response G(n)

• Implement the digital filter with the Generalised LTI architecture



What is it?

It sounds simple, but something does not fit in

What is it?

It sounds simple, but something does not fit in

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) = 255x(1) = 255

x(2) = 128x(3) = 128

x(4) = 255x(5) = 255

x(n)= [ 255,255,128,128,255,255]

Figure 14

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

3210

-1-2-3

x(n)

n

1 cycle

Figure 15




3210

-1-2-3

x(n)

n

2 cycle

Figure 16

Maximum number of cycles that can be represented in a sampling lattice.

3210

-1-2-3

x(n)

n


Figure 17


Maximum number of cycles = N/2

3210

-1-2-3

x(n)

n

Figure 18

Assuming that the sampling frequency is 1 Hertz or 2radians/second

The maximum frequency that can be represented is 1/2 Hertz or radians/second

Assuming that the sampling frequency is 1 Hertz or 2radians/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

x(n)

f

0 2N

2N

2N

2N

A typical spectrumA typical spectrum

Mirror Image

Mirror Image

Mirror Image

Figure 19

• 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 ‘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

X j x t e dtj t( ) ( )

x t X j e dj t( ) ( )

1

2

(5)

(6)

ForwardTransform

InverseTransform



• For digital sequence, we use Discrete Fourier Transform

(7)

(8)

ForwardTransform

InverseTransform

X x n e x n ekj nk N

n

Nj n

n

Nk( ) ( ) ( )/

2

0

1

0

1

x n X e X ekj nk N

k

N

kj n

k

Nk( ) ( ) ( )/

2

0

1

0

1

Nkk /2 with (9)

What are ‘n’ and ‘k’?What are ‘n’ and ‘k’?

ForwardTransform


n

Nj n

n

Nk( ) ( ) ( )/

2

0

1

0

1

n is an index to each sample of the waveform nx

3210

-1-2-3

nn=1 n=2 n=3

What are ‘n’ and ‘k’?What are ‘n’ and ‘k’?

ForwardTransform


n

Nj n

n

Nk( ) ( ) ( )/

2

0

1

0

1

k is an index to each frequency component in the spectrum kX

1

0

1

0

020

N

n

N

n

Nnj nxenxX )()()( /

Each frequency component is calculated with a value of ‘k’, e.g. The first (D.C.) component corresponds to k=0

1

0

21

0

121

N

n

NnjN

n

Nnj enxenxX // )()()(

The second component corresponds to k=1, and so on



• For digital filter design, the z Transform is often used as well. (See suplementary notes)

(10)Forward zTransform

n

nznxzX )()(

Unlike Fourier Transform, there is no simple inverse transform relation for z Transform.



• For digital filter design, the z Transform is often used as well

(10)Forward zTransform

n

nznxzX )()(

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’The basic meaning of ‘z’

r z = rej

Real

Img

Figure 20

A 2-D variable

Documents

Name: Dr. Peter Tsang Room: G6505 Ext: 7763 Email: [email protected]