Frequency Domain Representation of LTI Systems
In the context of LTI systems, Frequency Domain refers to the input domain of sinusoidal signals.
Consider an LTI system with imp. resp. h[n]
The input is e j0n
The output is
x[n] = e j0n
y[n] = ?
h[ n]
y[n ]
Due: Oct. 11, 2013
Middle East Technical University
Dept. of EEE
EE 430 Digital Signal Processing 2014 Fall
1) A discrete-time signal [] is obtained by sampling a continuous-time sinusoidal signal
() = 4 sin (20000 +
)
13
at a sampling rate of 3 kHz.
a)
Due:
Middle East Technical University
Dept. of EEE
EE 430 Digital Signal Processing 2014 Fall
No submission!
Computation of DFT
1) Consider a sequence whose length is a power-of-three. In this case, decimation-in-time can be applied by forming
subsequence
Due: Nov. 21, 2014
Middle East Technical University
Dept. of EEE
EE 430 Digital Signal Processing 2014 Fall
Submit for problems 2, 3, 6, 7, 8.
1) Let [] = [] + 3[ 1] + [ 2].
a. Plot [] and its periodic extension, [], for = 3 and = 5.
[] = [ ] = [() ]
=
b
Due: Dec. 1, 2014
Middle East Technical University
Dept. of EEE
EE 430 Digital Signal Processing 2014 Fall
Submit for problems 1,2,3,5,7,8
1) A function () is shown in the figure. Plot () =
= ( 8) for =50 and =20. Find the condition
on so that ( 8), , do
Due: Dec. 22, 2014
Middle East Technical University
Dept. of EEE
EE 430 Digital Signal Processing 2014 Fall
Submit for problems 0, 2, 5, 7, 11, 13, 15, 23, 25, 29
0) A photograph of what is in the title?
Transform Analysis of LTI Systems
1) The system fun
Due: Oct. 31, 2013
Middle East Technical University
Dept. of EEE
EE 430 Digital Signal Processing 2014 Fall
Submit for the first 9 problems. (29 in total)
1) The following difference equation for a causal LTI system is given,
yn
1
yn 1 xn xn 1 xn 2
2
a)
Unit - 1
Fall 2015
(part -1)
1
Signals - Introduction
Signal: Anything that carries some information can be called
as signal. A signal is also defined as any physical
quantity that varies with time, space or any other
independent variable or variables.
2
EE430 Digital Signal Processing
Signal processing is concerned with the representation, transformation and manipulation of
signals as well as the information they contain.
Signal Processing
Theory
Speech
Processing
Applications:
Image
Processing
Array
Pro
LTI Systems
CONVOLUTION
Linearity (and time-invariance) of a system, and representation of its input signal in terms of impulses
( x [n] =
x [ k ] [ n k ] ) can be combined to compute the output of that LTI system as the convolution of the
k =
input and
Linear Constant Coefficient Difference Equations
LTI systems man be represented by LCCDEs.
N
M
k =0
k =0
ak y [n k ] = bk x [ n k ]
y[n] + a1 y[n 1] + a 2 y[n 2] = b0 x[n] + b1 x[n 1] + b2 x[n 2]
x[n]
b0
+
y[n]
+
z-1
z-1
b1
+
+
a1
z-1
z-1
recursive part,
Name:
M.E.T.U. Electrical-Electronics Engineering Department
EE 430 Digital Signal Processing
QUIZ
Surname:
22.10.2010
Id No:
Attempt all (10) questions
8:40
Closed books and notes
1)
2)
3)
4)
5)
6)
7)
8)
9)
10)
Total:
1 ) A real LTI system has the follow
DFT
Discrete Fourier Transform
For a sequence x[n], its DTFT, X e j , is a function of a continuous variable ().
However, if x n is of finite length N, it can be considered as one period of its periodic extension
x n
x n k N .
k
Then, Fourier series re
SAMPLING
Continuous-time signals are commonly processed by using digital systems. Mobile devices, TV receivers and displays, radar,
sonar,
Widely used term is analog to digital, (A/D) conversion.
However, we will first consider continuous-time to discrete
Discrete-Time Signals
A sequence (ordering) of (real, complex) numbers, nth element is [ ]
.
x n
1
6
3 2
1
34
2
5
n
7
May have been obtained by sampling a continuous-time signal, i.e.,
Ex:
Let T = 0.001 sec = 1 msec.
, x 0.001 , x 0 , x 0.001 , x 0.002
We
z-Transform
Im cfw_ z
z-transform of a sequence x [ n ] is defined as
X (z ) =
x[n]z
n
,
z = re j
z C ,
n =
r
z
Re cfw_ z
Therefore X(z) is a complex valued function.
It takes complex values on the complex domain.
z n = r n e jn = r n (cos( n ) j sin (
Discrete-Time Systems
x n
Roughly stated,
A system is a transformation
A system is an input-output relationship
T
y n
SISO system
Ex: A delay system y n x n
Ex: y n ax n bx n 1
1 sample delay
a
b
N2
In general, y n ak x n k
+
N1
Ex: yn yn 1 xn
yn
n
x
Signal Processing
Signal
Processing
is
a
method
of
extracting information from the signal which
in turn depends on the type of signal and the
nature of information it carries.
1
What is a system?
A system is formally defined as an entity that manipulates