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DCSP-14 Jianfeng Feng Department of Computer Science Warwick Univ., UK

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How to deal with noise? How to transmit signals?

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Filter a filter is a device or process that removes from a signal some unwanted component or feature. Filtering is a class of signal processing, the defining feature of filters being the complete or partial suppression of some aspect of the signal. Most often, this means removing some frequencies and not others in order to suppress interfering signals and reduce background noise.

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Filter y(n) = a 0 x(n)+a 1 x(n-1)+…+a N x(n-N) +b 1 y(n-1)+…+b N y(n-N) Nth order filter

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Filter y(n) = a 0 x(n)+a 1 x(n-1)+…+a N x(n-N) +b 1 y(n-1)+…+b N y(n-N) Nth order filter signal

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Example y(n) = [ x(n)+ … x(n-N) ] / N In this case, when b i =0, it is an MA filter (averaging of input signals) In general it is called ARMA filter.

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Purpose To find the coefficient as and bs for certain purposes: For example, filter out noise, stop certain band signals, allow certain band signal to pass etc.

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Example w=y+.1*randn(length(y),1); for i=10:length(y) z(i)=(w(i)+w(i-1)+w(i-2)+w(i-3)+w(i-4)+w(i- 5)+w(i-6)+w(i-7)+w(i-8)+w(i-9))/10; end sound(z) sound(w)

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{y(n)} = a 0 {x(n)}+a 1 {x(n-1)}+…+a N {x(n-N)} +b 1 {y(n-1)}+…+b N {y(n-N)} Multiplying z -n on both size of the equation above where z is a complex number and summing over n Y(z) = a 0 X(z)+a 1 z -1 X(z)+…+a N z -N X(z) +b 1 z -1 Y(z)+…+b N z -N Y(z) Y(z)-( b 1 z -1 Y(z) +… + b N z -N Y(z)) = a 0 X(z) + a 1 z -1 X(z) +…+ a N z -N X(z)

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[1-b 1 z -1 -…- b N z -N ] Y(z) = [ a 0 + a 1 z -1 +… + a N z -N ] X(z) Y(z) ={ [ a 0 +a 1 z -1 +…+a N z -N ] / [1-b 1 z -1 -…-b N z -N ] } X(z) = H(z) X(z) H(z) is usually called transfer function: it characterizes the input output relationship of a filter

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Nonrecursive Filters When a filter is nonrecursive, its difference equation can be written y(n) = a 0 x(n) + a 1 x(n-1) +…+ a N x(n-N) Such filters are also called finite impulse response filters, for the obvious reason that their IR contain only finitely many nonzero terms. Correspondingly, the ZT of a nonrecursive filter can be written as a

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Block diagram

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z Sometime we use h=a

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Example

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Three types of representation Linear difference equation Block diagram Transfer function

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Trying to figure out how an FIR filter will behave, is not always so simple.

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Trying to figure out how an FIR filter will behave, is note always so simple. Another way of looking at it is through its frequency domain behaviors.

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Trying to figure out how an FIR filter will behave, is note always so simple. Another way of looking at it is through its frequency domain behaviors. We can make a start on this by examining the zeros of its transfer function H(z), i.e. those values of z for which H(z)=0 since H(z) is a polynomial of order N with real coefficients,

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Trying to figure out how an FIR filter will behave, is note always so simple. Another way of looking at it is through its frequency domain behaviors. We can make a start on this by examining the zeros of its transfer function H(z), i.e. those values of z for which H(z)=0 since H(z) is a polynomial of order N with real coefficients, It follows that the equation has N roots, which are either real or occur in complex conjugate pairs.

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We can express H(z) in terms of the roots by writing where in general z m = |z m | exp ( j arg [z m ] ) is the mth root, or zero of the transfer function. The zeros of a transfer function are usually denoted graphically in the complex z-plane by circles, as shown in the following Fig.

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We can express H(z) in terms of the roots by writing where in general z m = |z m | exp ( j arg [z m ] ) is the mth root, or zero of the transfer function.

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Recursive Filters Of the many filter transfer function which are not FIR, the most commonly use in DSP are the recursive filters, so called because their current output depends not only on the last N inputs but also on the last N outputs.

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