10. Z-Transform
10.1-10.4 General Principles of
Z-Transform
Z-Transform
Eigenfunction Property
x[n] = zn
y[n] = H(z)zn
h[n]
linear, time-invariant
H (z ) =
h [n ] z n
n =
1
p.2 of Chap.9
Chapters 3, 4, 5, 9, 10
Chap. 3
Chap. 4
Chap. 5
Im
Re
Chap. 9
Chap.
8. Communication Systems
Modulation: embedding an information-bearing
signal into a second signal
e.g.
x(t) : information-bearing signal
c(t) : carrier signal
y(t) = x(t)c(t) : modulated signal
purposes :
locate the signal on the right band of the spectr
6. Time/Frequency Characterization
of Signals/Systems
6.1 Magnitude and Phase for Signals
and Systems
x (t ), X ( jw )
h (t ), H ( jw )
y (t ), Y ( jw )
( ) h[n ], H (e ) y [n ], Y (e )
x [n ], X e jw
jw
jw
1
Signals
X ( jw ) = X ( jw ) e
X ( jw )
X ( jw
5. Discrete-time Fourier Transform
5.1 Discrete-time Fourier Transform
Representation for discrete-time signals
From Periodic to Aperiodic
Considering x[n], x[n]=0 for n > N2 or n < -N1
x
Construct ~ [n ] periodic with period N > N 1 + N 2 + 1
~ [n ] = x
4. Continuous-time Fourier Transform
From Fourier Series to Fourier Transform
Fourier Series : for periodic signal
x ( ) = x ( + T ), Tfundamental period
t
t
x (t ) =
a k e jkw t , w
k
0
=
as T increases, w 0
0
=
2
T
2
decreases
=
T
the envelope Ta k is
3 Fourier Series Representation of
Periodic Signals
3.2 Exponential/Sinusoidal Signals as
Building Blocks for Many Signals
1
Time/Frequency Domain Basis Sets
x[n]
Time domain
t
n
t
n
t
n
Frequency domain
0 1 2
0 1 2
t
n
t
n
composition
decomposition
2
Res
2. Linear Time-invariant Systems
2.1 Discrete-time Systems: the Convolution Sum
Representing an arbitrary signal as a sequence of
unit impulses
x[ n ] =
x[ k ] [ n k ]
k =
an unit impulse located at n = k on the index n
See Fig. 2.1, p.76 of text
u[n] =
VE216
Signals & Systems
Sung-Liang Chen
Summer 2013
Chapter 1
Signals and Systems
A Signal
z
A signal is a function of one or more variables,
which conveys information on the nature of some
physical phenomena.
z
Examples
z
f(t)
: a voice signal, a music s