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Dr. Ahmed El-Mahdy COMM 602: DSP
COMM 602: Digital Signal Processing
Prof. Ahmed El-Mahdy
Communications Department, German University
in Cairo.
e-mail: [email protected]
Dr. Ahmed El-Mahdy COMM 602: DSP
COMM 602: Digital Signal Processing
• Instructor: Prof. Ahmed El-Mahdy
• Office : C3.319
• Email: [email protected]
Dr. Ahmed El-Mahdy COMM 602: DSP
Text Book
J. Proakis, Digital Signal Processing:
Principles, Algorithms, and Applications”,
Communication Systems, 4th edition,
Prentice-Hall, 1996.
Dr. Ahmed El-Mahdy COMM 602: DSP
Grading • Assignments (2 Assignments best 2) 10%
• Practical Assignment 5%
• Quizzes (3 Quizzes best 2) 20%
(NO Compensation for the Quizzes)
• Midterm Exam 25%
• Final Exam 40%
Dr. Ahmed El-Mahdy COMM 602: DSP
Course Contents Subject No.
-Introduction to Digital Signal Processing. 1
-Linear Time Invariant Systems. 2
- The z-transform and its application to the LTI systems. 3
- Discrete Fourier Transform 4
- Fast Fourier Transform 5
- IIR Digital Filter Design. 6
- FIR Digital Filter Design. 7
Dr. Ahmed El-Mahdy COMM 602: DSP
COMM 602: Digital Signal Processing
Lecture 1
Introduction to Digital Signal Processing
Dr. Ahmed El-Mahdy COMM 602: DSP
What is DSP?
Dr. Ahmed El-Mahdy COMM 602: DSP
Advantages of Digital over Analog Signal Processing
• Flexibility: Digital programmable system, allows flexibility in
reconfiguring the digital signal processing operations simply by
changing the program. Reconfiguration of an analog system
usually implies a redesign of the hardware.
• Ex: Changing a filter from low pass to bandpass. In Digital: requires
changing the program, but in analog changing the components of the
circuit.
• Controlling the Accuracy: Digital systems provide much
better control of accuracy than analog systems by specifying
the accuracy requirements in A/D converter (number of
levels,... In analog systems, it is difficult to control the accuracy
because it is affected by other factors, for example, the circuit
components (resistors and capacitors) are affected by
temperature.
Dr. Ahmed El-Mahdy COMM 602: DSP
Advantages of Digital over Analog Signal Processing
• Storing: digital signals are easily stored on magnetic media
(disk), then the signal become transportable and can be
processed off-line.
• Cost: digital hardware is cheaper.
• Transmission: modern communication networks (internet &
LANS) use digital transmission.
Dr. Ahmed El-Mahdy COMM 602: DSP
Dr. Ahmed El-Mahdy COMM 602: DSP
Limitations of Digital Signal Processing
• Finite word length effect:
The use of limited number of bits (due to economical
considerations) affect the performance of the DSP systems.
Ex: Quantization with more levels requires more coding bits.
• Speed:
Analog signals with wide bandwidth require high sampling-
rate A/D converters to prevent Aliasing.
Dr. Ahmed El-Mahdy COMM 602: DSP
Analog to Digital (A/D) Conversion
Based on min error.
Dr. Ahmed El-Mahdy COMM 602: DSP
Sampling Theorem and Aliasing
Dr. Ahmed El-Mahdy COMM 602: DSP
Sampling Theorem and Aliasing (Contd.)
ms ff 2
Dr. Ahmed El-Mahdy COMM 602: DSP
Signal Classifications Signal is a representation of physical quantity or phenomenon
Deterministic Signals Random Signals
Time Domain Representation t is the independent variable
Frequency Domain Representation f is the independent variable
Representation
Continuous
Discrete
Continuous
Discrete
Dr. Ahmed El-Mahdy COMM 602: DSP
CONTINUOUS VERSUS DISCRETE TIME SIGNAL
Continuous Time Signal (CTS) Discrete Time Signal (DTS)
• the independent variable t is continuous.
• the signal values are defined for all t in
in the interval of interest.
• notation x(t).
• Example: volt or current.
• the independent variable t is discrete.
• it takes only a discrete values n.
• n is an integer.
• notation x(nTs)=x(n), Ts is the sampling period.
• it results from sampling of (CTS).
1
-1
5
-5
t
X(t)
0 2
-2
5
-5
n
X(n)
1
2.5
-2.5
- 1
0
Dr. Ahmed El-Mahdy COMM 602: DSP
Representation of Discrete Signals
Dr. Ahmed El-Mahdy COMM 602: DSP
Some Important Discrete Time Signals
1. Unit impulse (unit sample sequence)
00
01
n
nn
X(n)
. . . . . . . . . . . . 0 1 2 3 4 5 -1 -2 -3 -4 -5 6
n
. 1
Dr. Ahmed El-Mahdy COMM 602: DSP
Some Important Discrete Time Signals (Contd.)
2. Unit Step (unit sample sequence)
00
01
n
nnU
X(n)
. . . . . . . . . . . . 0 1 2 3 4 5 -1 -2 -3 -4 -5 6
n
. 1
Dr. Ahmed El-Mahdy COMM 602: DSP
Relationship between Unit impulse and unit step
Dr. Ahmed El-Mahdy COMM 602: DSP
Some Important Discrete Time Signals (Contd.)
3. Rectangular Signal
otherwise
Nnnx
0
01
X(n)
. . . . . . . . . . . . 0 1 2 -1 -2 -3 -4 -5 N
n
. 1
. ….
.
Dr. Ahmed El-Mahdy COMM 602: DSP
(Contd.)Some Important Discrete Time Signals
4. Real value exponential
aanx n
…….
a > 1 X(n)
. . . . . . . . . . . . 0 1 2 -1 -2 -3 -4 -5 n
. 1
. 3 4 5
X(n)
. . . . . . . . . . . . 0 1 2 -1 -2 -3 -4 -5 n
. 1
. 3 4 5
a < 1
Dr. Ahmed El-Mahdy COMM 602: DSP
Mathematical Formulas for the exponential signals
1a
Dr. Ahmed El-Mahdy COMM 602: DSP
(Contd.)Some Important Discrete Time Signals
5. Sinusoidal Signal
10 15 20 25 30
-1
-0.5
0
0.5
1
nwnxo
sin
n
x(n)
1
1 2 3 4 5……………………………..N
Dr. Ahmed El-Mahdy COMM 602: DSP
Operations on Signals
. Signal Addition1
• This is sample –to-sample addition and it is given by :
nxnxny 21
Or
nxnxny 21
nx1
nx2
nxnxny 21
+ + Adder
Dr. Ahmed El-Mahdy COMM 602: DSP
(Contd.)Operations on Signals
. Signal Multiplication2
• This is sample –to-sample multiplication given by :
nxnxny 21 .
Or
nxnxny 21 .
nx1
nx2
nxnxny 21 .
Multiplier
Dr. Ahmed El-Mahdy COMM 602: DSP
(Contd.)Operations on Signals
. Scaling3
• Each sample is multiplied by a scale or constant a
nxany
nx
a
. nxany
Dr. Ahmed El-Mahdy COMM 602: DSP
. Shifting:4
(Contd.)Operations on Signals
Dr. Ahmed El-Mahdy COMM 602: DSP
(Contd.)Operations on Signals
. Sample Summation5 • It adds all sample values of x(n) between n1 and n2
2
1
211 ..........1n
nn
nxnxnxnxny
. Sample Product6 • It multiplies all sample values of x(n) between n1 and n2
211 ..........12
1
nxnxnxnxnyn
nn
. Time Reversal7
• The signal y(n) = x (-n) is obtained by reflecting x (n) about n=0.
Dr. Ahmed El-Mahdy COMM 602: DSP
Example:
• Using basic building blocks introduced above, sketch the block diagram
representation of the discrete-time system described by input output
relation:
y(n)=0.25y(n-1)+0.5x(n)+0.5x(n-1)
where x(n) is the input and y(n) is the output of the system.
x(n)
z-1
0.5 z-1
0.25
y(n)
x(n-1)
y(n-1)
Dr. Ahmed El-Mahdy COMM 602: DSP
Energy and Power
2
n
nxE
3- For non-periodic discrete signal x(n)
12 lim
N
EP
Nav
4- For periodic discrete signal x(n) with period N
0
2
N
n
period nxE
N
EP
period
av
Dr. Ahmed El-Mahdy COMM 602: DSP
Periodic Signals
x(n) is periodic with period N, where N is a positive integer if
Nnxnx
N is called the fundamental period
. . . . . . . . . . . . . . . . . . . n
X(n)
N = 4
0 1 4 5
2 3
Dr. Ahmed El-Mahdy COMM 602: DSP
Even and Odd Signals x(n) is even if it is symmetric around y-axis nxnx
Examples
nUnx neither even nor odd 1-
2-
02
100
02
1
n
n
n
nxeven signal
3-
02
100
02
1
n
n
n
nx odd signal
x(n) is odd if it is symmetric around the origin nxnx
Dr. Ahmed El-Mahdy COMM 602: DSP
Discrete System or Digital system
a. Representation of a discrete system:
The discrete system is mathematically described as an operator or
transformation T [.]
• It takes an input sequence x(n), or the excitation, and transform it into another
sequence called the output sequence or the response y (n) .
T [.]
Processor
y(n)=T[x(n)]
input sequence
x(n)
Output sequence
y (n)
excitation response
b. Discrete System Analysis:
• It is the process of determining the response of that system to a given excitation
for any value of a
Basic System Properties
1. Linearity
Dr. Ahmed El-Mahdy COMM 602: DSP
Basic System Properties
Dr. Ahmed El-Mahdy COMM 602: DSP
Basic System Properties
Dr. Ahmed El-Mahdy COMM 602: DSP
Basic System Properties
2. Systems with memory and without memory
Memory less system
• If the output of the system dependents only on the present value of the input
signal. Otherwise, the system has memory.
System with memory
222 nxnxny
nxnxny 1
Dr. Ahmed El-Mahdy COMM 602: DSP
Basic System Properties (contd.)
3. CAUSALITY
Note: the causal system is a realizable one
A system is causal if the output at any time depends only on
values of the input at the present time and in the past. (the
system output does not anticipate the future values of the
input).
causalNonnxnxny
Causalnxnxny
]1[][][
]1[][][
Dr. Ahmed El-Mahdy COMM 602: DSP
Basic System Properties (contd.)
4. Time Invariance
• A system is time invariant if any delay in the input produces a
similar delay in the output.
• A specific input will produce the same output
independently of the time of application
. (1) y[n]=3x[n] …… Time invariant system
(2) y[n]=nx[n] …… Time variant system
Dr. Ahmed El-Mahdy COMM 602: DSP
• Example: Investigate the time invariance of the system:
y[n]=n x(n)
Solution
Let x1(n) be an arbitrary input to this system:
Then y1(n)=n x1(n)
is the corresponding output. Consider x2(n) =x1(n-no), then the
output corresponding to this input is:
y2(n)=n x2(n)=n x1(n-no) ……..(1)
Now we want to check whether y2[n]=y1[n-no] or not??
Then y1(n-no)=(n-no) x1(n-no) …………………(2)
which is not the same as y2(n). From (1) & (2), the system is time
variant
Dr. Ahmed El-Mahdy COMM 602: DSP
(contd.)Basic System Properties
The system is said to be stable if for a bounded input
there is a bounded output.
valuesfinite&veare,
,|][|,|][|
oi
oi
BandB
nBnynBnx
5. Stability
Dr. Ahmed El-Mahdy COMM 602: DSP
Example: • Consider a discrete time system whose input-output relation is
defined by:
Where r >1. Show that this system is unstable.
solution
for a bounded input x(t):
Then we find that:
With r>1, the output increases with increasing n. Then the system is
unstable.
,.....2,1,0],[][ nnxrny n
nallforMnx x ][
x
n
n
n
Mr
nxr
nxrny
][.
][][
Check the stability if r<1