Continuous-Time &
Discrete-Time Signal
Processing
ECE 3020 SIGNALS & SYSTEMS
DR. NU WAR
PROFESSOR
FACULTY OF COMPUTER SYSTEMS AND TECHNOLOGIES
MYANMAR INSTITUTE OF INFORMATION TECHNOLOGY
Contents
▪ Signal Transformation
▪ Basic System Properties
▪ Practices
Signal Transformation
SIGNAL TRANSFORMATION PROCESSES
Time shifting
Time reversal
Time scaling
Signal Transformation
A central concept in signal and system analysis is that of the transformation of a signal.
Examples
In an aircraft control system,
pilot’s actions are transformed by electrical and mechanical systems into changes in
aircraft velocity and heading;
In a high-fidelity audio system,
audio signal is transformed to enhance desirable characteristics, to remove recording
noise, or to balance the several components of the signal (e.g. treble and bass);
Time Shifting
For input 𝑥[𝑛] and 𝑥(𝑡), time shifted output are
𝑦 𝑛 =𝑥 𝑛−𝑛 ,
𝑦 𝑡 =𝑥 𝑡−𝑡 ,
where 𝑛0 and 𝑡0 are the number of samples to be shifted, respectively;
For 𝑛0 < 0 and 𝑡0 < 0, time-advance operation (left-shift); for 𝑛0 > 0 and 𝑡0 > 0, time-delay operation (rightshift);
*** E.g. radar, sonar, seismic signal processing
Signals at any two receivers at different locations result in a time shift.
Time Shifting
𝑛0 > 0, thus 𝑥[𝑛 − 𝑛0] is a delay version of 𝑥[𝑛]; (each point in 𝑥[𝑛] occurs later in 𝑥[𝑛 − 𝑛0])
Figure : A Discrete-Time Signal (Left) and its Delayed Version (Right)
Time Shifting
𝑥[𝑛] = {5, 6, 7, 8, 0, 9, 10, 11}, 𝑥[0] = 0.
For 𝑛0 = 3,
𝑥[𝑛 − 𝑛0] = ?
For 𝑛 = −1, 𝑥 [−1 − 3] = 5,
𝑛 = 0, 𝑥[0 − 3] = 6,
𝑛 = 1, 𝑥[1 − 3] = 7,
𝑛 = 2, 𝑥[2 − 3] = 8,
𝑛 = 3, 𝑥[3 − 3] = 0,
𝑛 = 4, 𝑥[4 − 3] = 9,
𝑛 = 5, 𝑥[5 − 3] = 10,
𝑛 = 6, 𝑥[6 − 3] = 11.
-4
-3
-2
𝑛
For 𝑛0 > 0, time-delay operation
(right-shift);
-1
0
1
2
3
4
5
6
𝑥[𝑛]
5
6
7
8
0
9
10
11
-
-
-
𝑥[𝑛 − 3]
-
-
-
5
6
7
8
0
9
10
11
Time Reversal
For 𝑥[𝑛], output is
For 𝑥(𝑡), output is
𝑦 𝑛 = 𝑥 −𝑛 .
𝑦 𝑡 = 𝑥 −𝑡 .
𝑥[−𝑛] and 𝑥(−𝑡) are obtained from 𝑥[𝑛] and 𝑥(𝑡) by a reflection at the origin at 𝑛 = 0 and 𝑡 = 0,
respectively.
*** E.g. if 𝑥(𝑡) is an audio tape, 𝑥(−𝑡) is the tape playing backward.
Figure : A Discrete-Time Signal (Left) and its Time Reversed Version (Right)
Time Scaling
For input 𝑥[𝑛], output is
For |𝛼| > 1, 𝑦[𝑛] is a linearly compressed version of 𝑥[𝑛].
For |𝛼| < 1, 𝑦[𝑛] is a linearly stretched version of 𝑥[𝑛].
Similarly for 𝑥(𝑡), output is
𝑦 𝑛 = 𝑥[𝛼𝑛].
𝑦(𝑡) = 𝑥(𝛼𝑡).
*** E.g. if 𝑥(𝑡) is an audio tape, 𝑥(2𝑡) is the tape playing at twice the speed, and 𝑥(𝑡/2) is the
tape playing at half the speed.
Time Scaling
A Continuous-Time
Signal
its Time Compressed
Version ,
Figure : Time Scaling of Continuous Time Signal
its Time Stretched Version
Transformation
Transformation of the signals
Time shifting
A signal is shifted in time to the left or to the right.
Time reversal
To enhance the desired quality of the signals
Flip a signal from the origin;
Time scaling
Time scaling on an audio clip causes perceptually compressed or stretched version of the
signal.
Assignment
▪ Let x[n]={1,2,3,4,5,6}. The signal x[n] is shown in Figure. Find and sketch the
following signals derived from x[n]:
x[n]
b)
6
5
4
3
2
1
-3 -2 -1 0 1
2 3
4
n
0
