DSPRevision I feng/dsp.html.

DSP—Revision I

http://www.dcs.warwick.ac.uk/~feng/dsp.html

Content• 1. Sequences and their representation • 2. Digital Filters• 3. Nonrecursive Filters• 4. Recursive Filters• 5. Frequency and digital filters• 6. Sampling and reconstruction• 7. Signal correlation and matched filters• 8. Dealing with noise• 9. Data compression• three weeks on FFT etc.• 10. Image feature extraction• 11. Image enhancement

2.3 Filters

General form

N

m

mm

N

m

mm

N

m

mm

N

m

mm

N

m

mm

N

m

mm

zb

za

zX

zYzH

zXzazYzb

nyDbnxDany

1

0

01

10

)1()(

)()(

)()()1(

)()()(

H is called the transfer function of the filter

4.3 Poles and zeros

kz

z

zH

m

nn

m

nn

1

1

N-1-

-N-1

)(

)(

b(N)zb(1)z-1

a(N)za(1)za(0))(

n is called poles, n is zeros

xzero

x

pole

A filter is fully determined by its poles and zeros

4.6 Three domains of representation

1. Time domain representation

Hx(n) y(n)

y(n)= b(1)y(n-1)+…+b(N)y(n-N) +a(0)x(n)+a(1)x(n-1)+…+a(N)x(n-N)

2. z--domain representation

))...((

))...((

)(...)1(

)(...)1()0()(

1

11

1

N

NNN

NN

zz

zzK

Nbzbz

NazazazH

K

1

x

X

1

2

2

3

3. frequency--domain representation

H()=H(z)|z=exp(j

|H()|

phase

ExampleTime: y(n)-y(n-1)+0.5y(n-2)=3x(n)-2x(n-1)

Z-domain: H(z)=(3-2z-1)/(1-z-1+0.5z-2)

= (3z2-2z)/(z2-z+0.5)

Zeros: 0, 2/3

Poles: 1/1.414 exp(j pi/4), 1/1.414 exp(-j pi/4)

It is BIBO stable

Frequency-domain:

Principle of filter design

1. We specify what the filter passes (the sign) and stop (the disturbance) in the frequency domain

2. Then we determine poles and zeros in the z-domain from the passband and the stopband

3. Finally, the filter is implemented recursively by the difference equation in the time domain

We can see that if the signal x(t) is bandlimited, in the sense that X(F)=0 for |F|>FB, for some frequency FB called the bandwidth of the signal, and we sample at a frequency Fs>2FB, then there is no overlapping between the repetitions on the frequency spectrum. In other words, if Fs>2FB,

Xs(F)=FsX(F) in the interval –Fs/2<F<Fs/2

And X(F) can be fully recovered from Xs(F). This is very important because it states that the signal x(t) can be fully recovered from its samples x(n)=x(nTs), provided we sample ‘fast enough’ (meaning Fs>2FB)

6.2.1 downsampling

(… x(-2) x(-1) x(0) x(1) x(2) ….)

(… y(-1) y(0) y(1) … )

)(2

1)(

2

1

))(12(2

1))(12(

2

1))(2(

))(12(2

1))(12(

2

1))(2(

))(2()2()()(

12122

12122

2

zXzX

zkxzkxzkx

zkxzkxzkx

zkxzkxzkyzY

k

k

k

k

k

k

k

k

k

k

k

k

k

k

k

k

k

k

6.2.2 Upsampling

(… x(-1) x(0) x(1) ….)

(… y(-2) 0 y(0) 0 y(2) … )

)(

))(()()()(

2

22

zX

zkxzkxzkyzYk

k

k

k

k

k

Example: Haar wavelet

Definition 1 (The Haar scaling function)   Let H be defined by H(t)=1, 0<t<=1, and 0 elsewhere

12,...1,0,...,1,0),2(2)(, jjjji ijitHtH

The index j refers to dilation and i refers to translation

Haar wavelet transform of a signal

...

3,23,22,22,21,21,20,20,2

1,11,10,10,1

0,00,00,00,0

GGfGGfGGfGGf

GGfGGf

GGfHHff

For any f as function in [0,1]. The decomposition is unique since {Gi,j} is orthogonal, and it forms a basis in L2

7.2 Correlation

)()()(

)()(

)()()(

nrnmxmx

mxnmx

mxmnhny

xxk

k

k

Its Z transform is

R(z)=X(z)X(z-1)

auto-correlation function

In general we have

cross-correlation function

R(z)=X(z)Y(z-1)

n

xy nmymxnR )()()(

8.1.3 Gaussian variables

Mean=, variance=

x=(…x(-2),x(-1),x(0),x(1),x(2),…) each of them is a Gaussian variable, then

is again a normal random variable with a mean and variance.

2

2

2

)(exp

2

1)(

x

xp

n

nxnhxh )()(

W(n)=x(n)+v(n)

Assume we know the signal sequence

x(1), x(2), ….,x(N)

How to design a filter so that we can detect the presence of the signal?

There are many ways to do it. The simplest and classical one is called

linear correlation detector.

9.3.1 Wiener Filter

Signal x

Received signal y=x+n

Minimize E(x-ay)^2 to find that

yEy

ENEyy

ENEx

Exayx

ENEx

Exa

2

22

22

2

22

2

Concentrating on transform coding

• Distributed multimedia

• JPEG, MPEG

• Using discrete cosine transform (DCT), a special case of DFT

10. Feature extraction

10.1 Matched filter

10.2 Gradient estimation

10.2 Local transforms

11. Enhancement

11.1 Contrast enhancement

11.2 Deblurring

11.3 Denoising

11.1 Contrast enhancement

• Histogram Equalization

Note how the image is extremely grey; it lacks detail since the

Example

Let x(i) be Gaussian random variables with mean zero and variance 1, and

y(n)= sin(n/2)x(n)+ sin((n-1)/2)x(n-1)+ sin((n-2)/2)x(n-2)+sin((n-3)/2)x(n-3)

Find Ey(n), E(y(n)-Ey(n))2, and the distribution of y(n)