Signal Processing - MATLAB Simulations for Radar Systems Design - page 638

Example:

Assume that the sampling signal p () is given b y

∞
∑

p () = δ tnT s ( – )

n = – ∞

Compute an expression for X s () .

Solution:

The signal p () is called the Comb function. Its exponential Fourier series

is

∞
∑

2π nt
T s

------------

1
T s

p () = ----- e

n = – ∞

It follows that

∞
∑

2π nt
T s

------------

x () 1
T s

----- e

x s () =

n = – ∞

Taking the Fourier transform of this equation yields

∞
∑

X s () 2π

2π n
T s

 

= ------ X ω – ---------- .

 

T s

n = – ∞

Before proceeding to the next section, we will establish the following nota-
tion: samples of the signal are denoted by and referred to as a dis-
crete time domain sequence, or simply a sequence. If the signal

x () x ()

x () is

ô ()

periodic, we will denote its sample by the periodic sequence .

13.10. The Z-Transform
The Z-transform is a transformation that maps samples of a discrete time
domain sequence into a new domain known as the z-domain. It is defined as

∞
∑

– n

Zx () { } = X () = x () z

(13.102)

n = – ∞