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

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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

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2.3 Filters General form H is called the transfer function of the filter

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4.3 Poles and zeros n is called poles, n is zeros

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x zero x pole A filter is fully determined by its poles and zeros

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4.6 Three domains of representation 1.Time domain representation H x(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)

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2. z--domain representation K 1 x X 1 2 2 3

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3. frequency--domain representation H( )=H(z)| z=exp(j |H( )| phase

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Example Time: 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 ) = (3z 2 -2z)/(z 2 -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:

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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

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We can see that if the signal x(t) is bandlimited, in the sense that X(F)=0 for |F|>F B, for some frequency F B called the bandwidth of the signal, and we sample at a frequency Fs>2F B, then there is no overlapping between the repetitions on the frequency spectrum. In other words, if F s >2F B, X s (F)=F s X(F) in the interval –F s /2<F<F s /2 And X(F) can be fully recovered from X s (F). This is very important because it states that the signal x(t) can be fully recovered from its samples x(n)=x(nT s ), provided we sample fast enough (meaning F s >2F B )

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6.2.1 downsampling (… x(-2) x(-1) x(0) x(1) x(2) ….) (… y(-1) y(0) y(1) … )

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6.2.2 Upsampling (… x(-1) x(0) x(1) ….) (… y(-2) 0 y(0) 0 y(2) … )

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Example: Haar wavelet Definition 1 (The Haar scaling function) Let H be defined by H(t)=1, 0<t<=1, and 0 elsewhere The index j refers to dilation and i refers to translation

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Haar wavelet transform of a signal For any f as function in [0,1]. The decomposition is unique since {G i,j } is orthogonal, and it forms a basis in L 2

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7.2 Correlation

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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 )

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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.

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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.

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9.3.1 Wiener Filter Signal x Received signal y=x+n Minimize E(x-ay)^2 to find that

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Concentrating on transform coding Distributed multimedia JPEG, MPEG Using discrete cosine transform (DCT), a special case of DFT

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10. Feature extraction 10.1 Matched filter 10.2 Gradient estimation 10.2 Local transforms

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11. Enhancement 11.1 Contrast enhancement 11.2 Deblurring 11.3 Denoising

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11.1 Contrast enhancement Histogram Equalization Note how the image is extremely grey; it lacks detail since the

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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)

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