1. Digital Signal Processing
بسم الله الرحمن الرحيم
Red Sea University
Department of Electrical And electronics Engineering
Fourth Year (sem 8)
Digital Signal Processing
Lecture 5
Lecturer : Elmustafa Sayed Ali Red Sea University - Engineering Faculty

2. Discrete Time Signals and Systems
Convolution Properties :
y(n) =x(n)*h(n) = 𝑘=−∞ ∞ 𝑥 𝑘 ∗ℎ(𝑛−𝑘 )
Cumulative
x(n)*h(n) = h(n)*x(n)
Associative
[x(n)*h1(n)]*h2(n) = x(n) *[h1(n)*h2(n)]
Distributive
x(n) *[h1(n)+h2(n)] = x(n)*h1(n)+x(n)*h2(n)
Lecturer : Elmustafa Sayed Ali Red Sea University - Engineering Faculty

3. Discrete Time Signals and Systems
Convolution methods:
Polynomial
Multiplication
Matrix
Polynomial:
X[n]= [ ] ; h[n] = [1 0 1]
x(n) = 1+2x+3 𝑥 𝑥 3
h(n)= 1+ 𝑥 2
y(n)= x(n)*h(n)
= 1+2x+3 𝑥 𝑥 3 + 𝑥 𝑥 3 +3 𝑥 4 +4 𝑥 5
= 1+2x+4 𝑥 2 +6 𝑥 3 +3 𝑥 4 +4 𝑥 5
y[n] = [ ]
Lecturer : Elmustafa Sayed Ali Red Sea University - Engineering Faculty

4. Discrete Time Signals and Systems
Multiplication :
x[n] = [ ] ; h[n]= [1 0 1]
(multiplying )
1 0 1
y[n]= [ ]
if there is any sum results after multiplication comes grater than 9 . Exp; 10 , 25 etc. It will be written as it came .
Lecturer : Elmustafa Sayed Ali Red Sea University - Engineering Faculty

5. Discrete Time Signals and Systems
Matrix :
x[n] = [ ] ; h[n]= [1 0 1] X 1 2 3 4
Multiplication
Summation
Lecturer : Elmustafa Sayed Ali Red Sea University - Engineering Faculty

6. Discrete Time Signals and Systems
1* (0+2) * (1+3)*(4+2)*(3+0)*4
Y[n]= [ ]
To dedicate the position of zero
X[n]=[ ] ; h(n)= [1 0 1]
0+0+2= 2 X 1 1 2 3 4
Lecturer : Elmustafa Sayed Ali Red Sea University - Engineering Faculty

7. Discrete Time Signals and Systems
Correlation:
Known as correlation between two signals is to measure the degree of which the two signals are similar.
The step of correlation are;
One of the signal shifted.
Then multiply the shifted signals with other one .
Two types of correlation are used :
Auto correlation
Cross correlation
Lecturer : Elmustafa Sayed Ali Red Sea University - Engineering Faculty

8. Discrete Time Signals and Systems
Auto correlation :
Represent correlation of signals with it self. Used to extract signal from noise .
Correlate signal with it self to identify the signal is it self or not .
Applications ;
Finger print system
Sound recognition
Cross correlation :
Correlate signal with another signals to compare input signal with the library of known signals stored in the system as used in radar.
Lecturer : Elmustafa Sayed Ali Red Sea University - Engineering Faculty

9. Discrete Time Signals and Systems
Example :
Given following two signals performs the correlation .
Sol:
Shifting right >>
y1= [ ]
y2(0)= [ ]
r0= [ ] = 2
y2(1)= [ ]
r1= [ ] = 0
y2(2)= [ ]
r2= [ ] = -1
Lecturer : Elmustafa Sayed Ali Red Sea University - Engineering Faculty

10. Discrete Time Signals and Systems
y1= [ ]
y2(4)= [ ]
r4= [ ] = 0
Shifting left <<
y2(-1)= [ ]
r-1= [ ] = 0
Lecturer : Elmustafa Sayed Ali Red Sea University - Engineering Faculty

11. Discrete Time Signals and Systems
y2(-3)= [ ]
r-3= [ ] = 0
The correlate signal obtained is;
rt= [ ] 2 -1 -1
Lecturer : Elmustafa Sayed Ali Red Sea University - Engineering Faculty

12. Discrete Time Signals and Systems
The original signal
The correlated signal
Correlation process gives a constructive interference .
Home work: constructive and distractive interferences ! 1 -1 -1 2 -1 -1
Lecturer : Elmustafa Sayed Ali Red Sea University - Engineering Faculty

13. Discrete Time Signals and Systems
Fourier Decomposition :
There are two types of signals maybe periodic or aperiodic;
Continuous signals
Discrete signals
Fourier decomposition is one of the two way to decompose the signals in signal processing , named by Joseph Fourier ( ) a French mathematician and physicist .
Fourier states that any continuous periodic signal could be represented as he sum o properly chosen sinusoidal waves.
Lecturer : Elmustafa Sayed Ali Red Sea University - Engineering Faculty

14. Discrete Time Signals and Systems
Components of sine and cosine waves are simpler than original signal because they have property than original signal (does not have). [sinusoidal fidelity].
Sinusoidal input to the system is guaranteed to produce a sinusoidal output. The amplitude and phase of signal may be changed but not frequency or shape.
Fourier transform divided into four categories .
Aperiodic continuous called Fourier transform. Eg; decaying exponential , Gaussian curve . The signal extended to both negative and positive infinity without repeating in periodic pattern .
Periodic continuous called Fourier series . Eg; sine , cosine , square waves . All these signals repeat it self in regular pattern from negative to positive infinity .
Lecturer : Elmustafa Sayed Ali Red Sea University - Engineering Faculty

15. Discrete Time Signals and Systems
Aperiodic discrete called discrete time Fourier transform. Eg; this signals are only defined in discrete points between positive and negative infinity , and not repeated themselves in periodic fashion .
Periodic discrete called discrete Fourier series or discrete Fourier transform . Signals defined in discrete points between positive and negative infinity and repeat themselves.
Lecturer : Elmustafa Sayed Ali Red Sea University - Engineering Faculty

16. Thank To All for Listening
The End
Thank To All for Listening
Any Questions
Lecturer : Elmustafa Sayed Ali Red Sea University - Engineering Faculty