15
Lecture 4
Properties of ROC
1) ROC is a ring o r1 < z < r2
2) The Fourier Transform of x(n) converges if and only if ROC of X(z) includes the unit circle.
(Remember, z = rej and if z = 1 then r = 1 and then X ( z ) = X (e j ) = x(n ) e j n =
+
Fouri
21
Lecture 5
Calculating the Inverse ZTransform
x(n ) =
1
n 1
X ( z ) z dz Three methods to calculate it:
2j
1) Direct Method by contour integration
2) Expansion into a series of terms z/z1
3) Partial Fraction expansion and lookup table.
CauchyResidu
14
Lecture 3
Implementation of DiscreteTime Systems
Lets start with 1st order system:
y(n) = a1y(n 1) +box(n) + b1 x(n 1) or
M
N
k =0
k =0
ak y(n k ) = bk x(n k )
This can be considered as v(n ) = bo x(n ) + b1 x(n 1) and
y (n ) = a1 y (n 1) + v(n )
bo
31
Lecture 6 Chapter 4
Frequency Analysis of Signals and Systems
Continuous Signals and DiscreteTime Signals
Periodic
Aperiodic
Starting with periodic CT signals:
Recall that a linear combination of harmonically related complex exponentials of the form
6
Lecture 2 (Chapter 2) DiscreteTime Signal and Systems
Classification of Signals
1. Finite duration
x(n) =0 n > N
Infinite duration
2. RightLeft sided
x(n ) = 0
x(n ) = 0
n < N 1 rightsided
n > N 2 leftsided
Some Elementary DiscreteTime Signals

1
Fourier Analysis Using the DFT
Fourier Analysis of Signals Using DFT
One major application of the DFT: analyze signals
Lets analyze frequency content of a continuoustime signal
Steps to analyze signal with DFT
Remove highfrequencies to prevent aliasi
DiscreteTime Fourier Transform Properties
Absolute and Square Summability
Absolute summability is sufficient condition for DTFT
Some sequences may not be absolute summable but only
square summable
xn
2
n
To represent square summable sequences with DT
DiscreteTime Signals and Systems
DiscreteTime Signals: Sequences
Discretetime signals are represented by sequence of numbers
The nth number in the sequence is represented with x[n]
Often times sequences are obtained by sampling of
continuoustime si
Fast Fourier Transforms
Discrete Fourier Transform
The DFT pair was given as
N 1
Xk x[n]e j 2 / Nkn
n 0
1 N 1
x[n] Xk e j 2 / Nkn
N k 0
Baseline for computational complexity:
Each DFT coefficient requires
N complex multiplications
N1 complex additio
Linear TimeInvariant Systems
LinearTime Invariant System
Special importance for their mathematical tractability
Most signal processing applications involve LTI systems
LTI system can be completely characterized by their impulse
response
Tcfw_.
[nk]
DiscreteTime Fourier Transform
Frequency Response
The frequency response defines a systems output
for complex exponential at all frequencies
If input signals can be represented as a sum of complex
exponentials
xn k e j kn
k
we can determine the outpu
Changing the Sampling Rate
Changing the Sampling Rate
A continuoustime signal can be represented by its samples as
xn x c nT
We can use bandlimited interpolation to go back to the
continuoustime signal from its samples
Some applications require us t
Frequency Response of Rational Systems
Frequency Response of Rational System Functions
DTFT of a stable and LTI rational system function
M
He
j
bk e
M
1 c e
j k
k 0
N
j k
a
e
k
b
0
a0
k 0
j
k
k 1
N
1 d e
j
k
k 1
Magnitude Response
M
H e
The Discrete Fourier Transform
Sampling the Fourier Transform
Consider an aperiodic sequence with a Fourier transform
x[n] DTFT
X e j
Assume that a sequence is obtained by sampling the DTFT
~
X k X e j
X e j 2 / N k
2 / N k
Since the DTFT is p
Relationship between Magnitude and Phase
Relation between Magnitude and Phase
For general LTI system
Knowledge about magnitude doesnt provide any information
about phase
Knowledge about phase doesnt provide any information about
magnitude
For linear c