# Chapter 4 The Z-Transform and Fourier Transforms of Discrete-Time (1)

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```					To help you do the revision, here is the compilation of a list of key theoretic points for you to focus on. Try to ask yourself whether you have acquired the required knowledge.

Week 1-2: Discrete-Time Signals and Systems
Discrete time signals
Three Fundamental Sequences
1. Unit sample
1 n  0 0 otherwise

 ( n)  
2. Unit step

1 n  0 u ( n)   0 otherwise Relationship between u(n) and δ(n):
u ( n) 
n  

  (k )

n

δ(n) = u(n) - u(n-1) 3. Exponential sequences x(n) = an a may be real or complex number.

a  e jw0

is quite useful.

Classifications of Discrete Time Signals
1. Signal Duration Finite length sequence – equals zero beyond [N1,N2] Infinite length sequence – ex. Unit step  Right sided sequence: zero for n less than n0  Left sided sequence: zero for n bigger than n0  Two sided sequence 2. Periodic and Aperiodic sequences A signal x(n) is said to be periodic if, for some positive real integer N: x(n) = x(n+N) Fundamental period – N is smallest integer of the last equation. 3. Symmetric Sequences A real valued signal is said to be even if, for all n: x(n) = x(-n) Whereas a signal is said to be odd if, for all n: x(n) =- x(-n) Any signal can be decomposed as a combination of even and odd signal: x(n) = xe(n) + xo(n) xe(n) = ½ [(x(n) + x(-n) ] xo(n) = ½ [(x(n) - x(-n) ] Complex value sequence: It is said to be conjugate symmetric if, for all n x(n) = x*(-n)

It is said to be conjugate asymmetric if, for all n x(n) = - x*(-n) 4. Signal Decomposition:
x ( n) 
k  

 x(k ) (n  k )



Discrete-time Systems and properties
A discrete-time system is a mathematical operator or mapping that transforms one signal ( the input) into another signal ( the output) by means of a fixed set of rules or operation. y(n)=T[x(n)]

x(n)

T[.]

1. Memory-less system Definition: A system is said to be memoryless if the output at any time n=n0 depends only on the input at time n=n0. Ex: y(n) = x2(n) Y(n) = x(n)+x(n-1) 2. Additive systems: T[x1(n) + x2(n)] = T[x1(n)] + T[x2(n)] 3. Homogeneity: T[cx(n)] =c T[x(n)] 4. Linear system: T[a1x1(n) + a2x2(n)] =a1 T[x1(n)] + a2T[x2(n)] h(n) = T[δ(n)] hk(n) = T[δ(n-k)]
y ( n) 
k  

 x(k )h



k

( n)

5. Shift Invariant System: For y(n)=T[x(n), the system is said to be shift invariant if, for any delay n0, the response to x(n-n0) is y(n-n0). 6. LSI ( Linear Shift Invariant) System: For LSI : hk(n) = h(n-k) For LSI system, any input x(n) will have output:
y ( n) 

k  

 x(k )h(n  k ) = x(n)*h(n)



7. Causality

A system is said to be causal if, for any n0 the response of the system at time n0 depends only on the input to time n= n0. 8. Stability A sytem is said to be stable in the bounded input-bounded output sense if, for any input that is bounded x(n)  A   , the output will be bounded,

y ( n)  B  

Convolution Sums
k  

 x(k )h(n  k ) = x(n)*h(n)



Properties:
Commutative Property: x(n)*h(n) = h(n)*x(n) Associative Property: {x(n)*h1(n)}*h2(n) =x(n)*{h1(n)*h2(n) } Distributive Property: x(n)*{h1(n)+h2(n)} =x(n)*h1(n)+x(n)*h2(n)

Performing Convolution
Direct Evaluation Graphic Approach Sliding Rule Method.

Difference Equations
LCCDE – Linear Constant Coefficient Difference Equation:

y ( n)   b ( k ) x ( n  k )   a ( k ) y ( n  k )
k 0 k 1

q

p

If the system have one or more a(k) that are nonzero, the difference equation is said to be recursive. If all of the coefficients a(k) are equal to zero, the difference equation is said to be non recursive. Difference equation provide a method for computing the response of a system, y(n), to an arbitrary input x(n). Approaches to solve LCCDE:  Classical approach of finding homogeneous and particular solution.

Week 3-4: DTFT
Frequency Response
1. For a LSI system, if input signal x(n)  e jn    n   , where  is a constant, the output of the system y (n)  h(n)  x(n)  H (e j )e jn

where
H (e j ) 
k  

 h ( k )e



 jk

which is called the frequency response. 2. H (e j )  H R (e j )  jH I (e j ) or H (e j )  H (e j ) e jh ( ) where magnitude phase group delay
2 H (e j )  H R (e j )  H I2 (e j ) 2

H I ( e j ) H R ( e j ) d ( )  h ( )   h d

 h ( )  tan 1

3. Properties (1) Periodicity The frequency response is a complex-valued function of  and is periodic with a period 2 . (2) Symmetry If h(n) is real-valued, the frequency response is a conjugate symmetric function of frequency: H (e  j )  H  (e j ) 4. Inverting the frequency response The unit sample response may be recovered by integration 1  j jn h(n)   H (e )e d 2

The Discrete-Time Fourier Transform
1. Definition
X ( e j ) 
n   

 x ( n)e

 jn

.

Thus, the frequency response of a LSI system, H (e j ) , is the DTFT of the unit sample response, h(n) . The existence requires that x (n) be absolutely summable:
n  

 x ( n)  S  



2. Properties Property Linearity Shift Time-reversal Sequence ax(n)  by (n)
x ( n  n0 )
x (  n)

DTFT aX (e )  bY (e j )
j

e  jn0 X (e j )

X ( e  j )

Modulation Convolution Conjugation Derivative Multiplication 3. Applications

e jn 0 x(n) x ( n)  y ( n) x  (n)
nx(n) x ( n) y ( n)

X (e j (  0 ) )
X (e j )Y (e j )

X  (e  j )

j

dX (e j ) d
j

1 2

  X (e




)Y (e j (  ) ) d

Interconnection of Systems
1. Series (cascade) x(n) h1(n) h2(n) y(n)

h(n)  h1 (n)  h2 (n) H (e j )  H 1 (e j ) H 2 (e j )
log H (e j )  log H 1 (e j )  log H 2 (e j )

 ( )  1 ( )  2 ( )  ( )   1 ( )   2 ( )
2. Parallel h1(n) x(n) + y(n)

+
h2(n)

+

h(n)  h1 (n)  h2 (n) H (e j )  H 1 (e j )  H 2 (e j )

Week 5- 6: z - Transform
Definition of Z-Transform
1. Definition The z-transform of a discrete-time signal x(n) is defined by

X ( z) 
j

n  

 x ( n) z



n

where z  re is a complex variable. The values of z for which the sum converges define a region in z-plane referred to as the region of convergence (ROC). 2. Notation If x(n) has a z-transform X(z), we write x ( n)  Z X ( z )  3. ROC ROC is determined by the range of values of r for which
n  

 x ( n) r



n



Table 4-1 Common z-Transform Paris Sequence  (n) z-Transform 1 1 1  z 1 1 1  z 1 z 1 (1  z 1 ) 2 Region of Convergence all z

 n u (n)
  n u (n  1)

z 
z 

n n u (n)
 n u (n  1)
n

z  z  z 1 z 1

z 1 (1  z 1 ) 2
1  (cos  0 ) z 1 1  2(cos  0 ) z 1  z  2 1  (cos  0 ) z 1 1  2(cos  0 ) z 1  z  2

cos(n 0 )u (n) sin( n 0 )u (n)

4. Complex z-plane z  Re( z )  j Im( z )  re j Unit circle:

z 1

Im(z) Unit circle



Re(z)

Properties of Z-Transform
1. Linearity If x(n) has a z-transform X(z) with a region of convergence Rx, and if y(n) has a ztransform Y(z) with a region of convergence Ry, w(n)  ax(n)  by (n) Z W ( z )  aX ( z )  bY ( z )  and the ROC of W(z) will include the intersection of Rx and Ry, that is, Rw contains Rx  R y . 2. Shifting property If x(n) has a z-transform X(z), x(n  n0 ) Z z n0 X ( z) .  3. Time reversal If x(n) has a z-transform X(z) with a region of convergence Rx that is the annulus   z   , the z-transform of the time-reversed sequence x(-n) is
x(n) Z X ( z 1 ) 

and has a region of convergence 1   z  1  , which is denoted by 1 R x . 4. Multiplication by an exponential If a sequence x(n) is multiplied by a complex exponential  n ,  n x(n) Z X ( 1 z ) .  5. Convolution theorm If x(n) has a z-transform X(z) with a region of convergence Rx, and if h(n) has a ztransform H(z) with a region of convergence Rh, y ( n )  x ( n)  h( n)  Z Y ( z )  X ( z ) H ( z ) .  The ROC of Y(z) will include the intersection of Rx and Rh, that is,

Ry contains R x  Rh . With x(n), y(n), and h(n) denoting the input, output, and unit-sample response, respectively, and X(z), Y(x), and H(z) their z-transforms. The z-transform of the unitsample response is often referred to as the system function. 6. Conjugation If X(z) is the z-transform of x(n), the z-transform of the complex conjugate of x(n) is x  (n) Z X  ( z  ) .  7. Derivative If X(z) is the z-transform of x(n), the z-transform of n k x(n) is dX ( z ) . nx(n) Z  z  dz 8. Initial value theorm If X(z) is the z-transform of x(n) and x(n) is equal to zero for n<0, the initial value, x(0), maybe be found from X(z) as follows: x(0)  lim X ( z ) .
z 

Inverse Z-Transform
If X(z) is the z-transform of x(n), the inverse z-transform is given by the contour integral 1 x ( n)  X ( z ) z n1dz , 2j C where C is a counterclockwise closed contour in the region of convergence of X(z) and encircling the origin of the z-plane.

Week 7-10 : DFS & DFT
Discrete Fourier Series
1. Definition x Let ~(n) be a periodic sequence with a period N : ~(n)  ~(n  N ) . x x It can be expressed in terms of a discrete Fourier series (DFS) as follows: N 1 ~ ~ ( n)  1 x  X (k )e j 2nk / N N k 0 N 1 ~ X (k )   ~ (n)e  j 2nk / N x
n 0

Note that the DFS coefficients are periodic with a period N : ~ ~ X ( k  N )  X (k ) . We write DFS as

Define WN  e  j 2 / N , then

DFS ~ ~(n) x  X (k )

N 1 ~ ~ ( n)  1 x  X (k )WNnk N k 0 N 1 ~ nk X (k )   ~(n)WN x n 0

2. Properties (a) Linearity Let ~1 (n) and ~2 (n) are periodic with a period N , x x DFS ~ ~ ~ (n)  ~ (n) x1 x 2  X 1 (k )  X 2 (k ) (b) Shift y x (1) ~(n)  ~(n  n0 ) ~ kn ~ Y (k )  WN 0 X (k ) nk (2) ~(n)  WN 0 ~(n) y x ~ ~ Y (k )  X (k  k 0 ) (c) Periodic convolution ~ ~ If h (n) and ~(n) are periodic with a period N with DFS coefficients H (k ) x ~ and X (k ) , the sequence with DFS cofficients ~ ~ ~ Y (k )  H ( k ) X (k ) ~ is formed by periodically convolving h (n) with ~(n) as follows: x
~ ~ ( n)  h ( k ) ~ ( n  k ) y  x
k 0 N 1

Notationally, the periodic convolution of two sequences is written as ~ ~(n)  h (n) O ~(n) y * x

Discrete Fourier Transform
1. Definition Let x (n) be a finite-length sequence of length N that is equal to zero outside the interval [0, N  1] . A periodic sequence ~(n)  x(n mod N )  x((n)) x N The sequence X (k ) is called the N-point DFT of x (n)

X (k )   x(n)e  j 2nk / N
n 0

N 1

x(n)  X (k )
0k  N

DFT

X (k )  X (e j )
2. Properties (a) Linearity

  2k / N

 X ( z) z exp{2k / N }

ax1 (n)  bx2 (n)  aX 1 (k )  bX 2 (k )
(b) Symmetry If x (n) is real-valued,

DFT

X (k )  X  ((k ))  X  (( N  k )) N , and if x (n) is imaginary, X (k )   X  ((k ))   X  (( N  k )) N
(c) Circular shift The circular shift of a sequence x (n) is defined as follows: x((n  n0 )) N RN (n)  ~(n  n0 ) RN (n) x where n0 is the amount of the shift and RN (n) is a rectangular window:
1 R N ( n)   0 0n N else
DFT

n x((n  n0 )) N RN (n) WN 0k X (k ) nk WN 0 x(n)  X ((k  k 0 )) N DFT

3. Circular convolution Let h(n) and x (n) be finite-length sequences of length N with N -point DFTs H (k ) and X (k ) , respectively. The N-point circular convolution y (n) of h(n) and x (n) has a DFT equal to the product Y (k )  H (k ) X (k ) .
 N 1 ~   N 1 ~  y ( n)   h ( k ) ~ ( n  k )  R N ( n)   h ( n  k ) ~ ( k )  R N ( n) x x  k 0   k 0  y (n)  h(n) x (n) = x (n) h(n) N N ~ x y (n)  h(n) N x (n) = h (n) * ~(n) RN (n)





4. Circular versus linear convolution y ( n )  x ( n )  h( n )
   x(n) N h(n)    y (n  kN ) R N (n) k    5. Linear convolution using the DFT (a) Pad the sequences h(n) and x (n) with zeros so that they are of length N  N1  N 2  1. (b) Find the N -point DFTs of h(n) and x (n) . (c) Multiply the DFTs to form the product Y (k )  H (k ) X (k ) . (d) Find the inverse DFT of Y (k ).

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