1. DCSP-12 Jianfeng Feng Department of Computer Science Warwick Univ., UK Jianfeng.feng@warwick.ac.uk http://www.dcs.warwick.ac.uk/~feng/dsp.html

2. Signal Representation

4. 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).

5. 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.

6. 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.

7. 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 )

8. 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)

9. 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.

10. 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

11. 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.

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

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