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DCSP-12 Jianfeng Feng Department of Computer Science Warwick Univ., UK

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Signal Representation

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Sequences and their representation A sequence is an infinite series of real numbers { x(n) }, which is written { x(n) } = {…, x(-1),x(0),x(1),x(2), …,x(n), … } This can be used to represent a sampled signal, i.e. x(n) = x(nT), where x(t) is the original (continuous) function of time. Sometimes sequence elements are subscripted, x n being used in place of x(n).

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The most basic tool in DCSP is the Z transform (ZT), which is related to the generating function used in the analysis of series. In mathematics and signal processing, the Z- transform converts a discrete time domain signal, which is a sequence of real numbers, into a complex domain representation.

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The ZT of { x(n) } is X(z)= x(n) z -n where the variable z is a complex number in general and the summation The first point to note about ZT's is that some sequences have simple rational ZT's.

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Example { x(n), n=1,2,…,} = {1, r, r 2, …..}, x(n)=0, n<0 has ZT X(z) = r n z -n which can be simplified if r/|z|<1 to give X( z ) = 1 / (1-r z -1 )

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The ZT's main property concerns the effects of a shift of the original sequence. Consider the ZT Y(z)= x(n) z -(m+n) which can be written Y(z) = z -m X(z)

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This obviously corresponds to the sequence y(n) = { x(n-m) } Now if we add sequences { a(n) }, { b(n) }, we get a sequence { c(n) }= { …,a(-1)+b(-1),a(0)+b(0), a(1)+b(1), … } with ZT C(z)= ( a(n) + b(n) ) z -n = A(z)+B(z) which is just the sum of the ZT's of {a(n)}, {b(n)}, i.e. the ZT is linear.

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Now consider the product of two ZT's C(z) = A(z) B(z) for example. The question arises as to what sequence this represents. If we write it out in full

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Sequences { c(n) } of this form are called the convolution of the two component sequences a(n), b(n), and are sometimes written as c(n)=a(n)*b(n) Convolution describes the operation of a digital filter, as we shall see in due course. The fundamental reason why we use ZT is that convolution is reduced to multiplication and this is a consequence of the even more basic shift property expressed in the equation above.

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Example (Discrete time unit impulse) The unit impulse (n) is the most elementary signal, and it provides the simplest expansion. It is defined as

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Any discrete signal can be expanded into the superposition of elementary shifted impulse, each one representing each of the samples. This is expressed as

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