Introduction to signals and systems

1. Signals and Systems
Sarun Soman
Asst: Professor
Manipal Institute of Technology
Manipal

2. References
1. Haykin S; Signals and Systems, Wiley, 1999
2. Oppenheium, Willisky and Nawab; Signals and
Systems (2e), PUI, 1997.
Prof: Sarun Soman, MIT, Manipal

3. What is a signal?
A signal is formally defined as a function of one or more
variable that conveys information on the nature of a
physical phenomenon.
One dimensional signal: function of a single variable.
Eg. Speech signal
Two dimensional signal: functions of two variables.
Eg. Image
Prof: Sarun Soman, MIT, Manipal

4. What is a system?
A system is formally defined as an entity that
manipulates one or more signals to accomplish a
function, thereby yielding new signals.
voltage
MOTOR
Mechanical work
i/p o/p
System
Prof: Sarun Soman, MIT, Manipal

5. Application Areas
• Control
• Communications
• Signal Processing
Prof: Sarun Soman, MIT, Manipal

6. Control Applications
• Industrial control and automation (Control the
velocity or position of an object)
• Examples: Controlling the position of a valve
or shaft of a motor
• Important Tools:
– Time-domain solution of differential equations
– Transfer function (Laplace Transform)
– Stability
Prof: Sarun Soman, MIT, Manipal

7. Communication Applications
• Transmission of information (signal) over a
channel
• The channel may be free space, coaxial cable,
fiber optic cable
• A key component of transmission: Modulation
(Analog and Digital Communication)
Prof: Sarun Soman, MIT, Manipal

8. Signal Processing Applications
• Signal processing: Application of algorithms to
modify signals in a way to make them more useful.
• Goals:
– Efficient and reliable transmission, storage and display of
information
– Information extraction and enhancement
• Examples:
– Speech and audio processing
– Multimedia processing (image and video)
– Underwater acoustic
– Biological signal analysis
Prof: Sarun Soman, MIT, Manipal

9. Classification of signals
1. Continuous Time (CT)and Discrete Time (DT)signals
2. Even and odd signals
3. Periodic signals and non periodic signals
4. Deterministic signals and random signals
5. Energy and Power signals
Prof: Sarun Soman, MIT, Manipal

10. CT and DT signals
CT signal
A signal is said to be continuous time signal if it is defined
for all time t.
Prof: Sarun Soman, MIT, Manipal

11. CT and DT signals
DT signals
A discrete time signal is defined only at discrete
instants of time.
‫ݐ‬ = ݊ܶ௦
‫ݔ‬ ݊ = ‫ݔ‬ ݊ܶ௦ , ݊ = 0, ±1, ±2, ±3
ܶ௦ ݅‫݀݋݅ݎ݁݌ ݈݃݊݅݌݉ܽݏ ݄݁ݐ ݏ‬
Prof: Sarun Soman, MIT, Manipal

12. Even and Odd signals
Even signals:
‫ݔ‬ −‫ݐ‬ = ‫ݔ‬ ‫ݐ‬ ݂‫ݐ ݈݈ܽ ݎ݋‬
‫ݔ‬ −݊ = ‫ݔ‬ ݊ ݂‫݊ ݈݈ܽ ݎ݋‬
‫)݊(ݔ‬
Symmetric about vertical axis
Prof: Sarun Soman, MIT, Manipal

13. Even and Odd signal
Odd signals:
‫ݔ‬ −‫ݐ‬ = −‫ݔ‬ ‫ݐ‬ ݂‫ݐ ݈݈ܽ ݎ݋‬
‫ݔ‬ −݊ = −‫ݔ‬ ݊ ݂‫݊ ݈݈ܽ ݎ݋‬
‫)݊(ݔ‬
Antisymmetric about origin
Prof: Sarun Soman, MIT, Manipal

14. Example(even and odd signal)
Consider the signal x ‫ݐ‬ = ቊ
sin
గ௧
்
, −ܶ ≤ ‫ݐ‬ ≤ ܶ
0, ‫݁ݏ݅ݓݎ݄݁ݐ݋‬
.
Is the signal x(t) an even or an odd function of time t?
Ans:
Replacing t with –t yields.
x −‫ݐ‬ = ൝
sin
−ߨ‫ݐ‬
ܶ
, −ܶ ≤ ‫ݐ‬ ≤ ܶ
0, ‫݁ݏ݅ݓݎ݄݁ݐ݋‬
= ቊ
− sin
గ௧
்
, −ܶ ≤ ‫ݐ‬ ≤ ܶ
0, ‫݁ݏ݅ݓݎ݄݁ݐ݋‬
= −‫ݔ‬ ‫ݐ‬ ݂‫ݐ ݈݈ܽ ݎ݋‬
x(t) is an odd signal
Prof: Sarun Soman, MIT, Manipal

15. Even-Odd decomposition of a signal
Consider an arbitrary signal x(t)
Let ‫ݔ‬ ‫ݐ‬ = ‫ݔ‬௘ ‫ݐ‬ + ‫ݔ‬଴(‫)ݐ‬ (1)
‫ݔ‬௘ −‫ݐ‬ = ‫ݔ‬௘(‫)ݐ‬
‫ݔ‬଴ −‫ݐ‬ = −‫ݔ‬଴(‫)ݐ‬
Substitute t = -t in (1)
‫ݔ‬ −‫ݐ‬ = ‫ݔ‬௘ −‫ݐ‬ + ‫ݔ‬଴(−‫)ݐ‬
= ‫ݔ‬௘ ‫ݐ‬ − ‫ݔ‬଴(‫)ݐ‬ (2)
Solving for ‫ݔ‬௘ ‫ݐ‬ [(1) + (2)]
‫ݔ‬௘ ‫ݐ‬ =
ଵ
ଶ
‫ݔ‬ ‫ݐ‬ + ‫ݔ‬ −‫ݐ‬
Solving for ‫ݔ‬଴(‫)ݐ‬ [(1) – (2)]
‫ݔ‬଴ ‫ݐ‬ =
ଵ
ଶ
‫ݔ‬ ‫ݐ‬ − ‫ݔ‬ −‫ݐ‬
Prof: Sarun Soman, MIT, Manipal

16. Even-Odd decomposition of a signal
Hyperbolic sine and cosine function
‫ݔ݄݊݅ݏ‬ =
݁௫
− ݁ି௫
2
ܿ‫ݔ݄ݏ݋‬ =
௘ೣା௘షೣ
ଶ
Odd and even functions
sinh −‫ݔ‬ = −‫ݔ݄݊݅ݏ‬
cosh −‫ݔ‬ = ܿ‫ݔ݄ݏ݋‬
Prof: Sarun Soman, MIT, Manipal

17. Even-Odd decomposition of a signal
Find the even and odd components of the signal
‫ݔ‬ ‫ݐ‬ = ݁ିଶ௧
cos ‫ݐ‬
Ans:
Replacing t with –t yields
‫ݔ‬ −‫ݐ‬ = ݁ଶ௧
cos(−‫)ݐ‬
= ݁ଶ௧ cos(‫)ݐ‬
Even component
‫ݔ‬௘ ‫ݐ‬ =
ଵ
ଶ
‫ݔ‬ ‫ݐ‬ + ‫ݔ‬ −‫ݐ‬
=
ଵ
ଶ
݁ିଶ௧
cos ‫ݐ‬ + ݁ଶ௧
cos(‫)ݐ‬
= cos ‫ݐ‬
௘షమ೟ା௘మ೟
ଶ
= cosh 2‫ݐ‬ cos ‫ݐ‬
Prof: Sarun Soman, MIT, Manipal

18. Even-Odd decomposition of a signal
Odd component
‫ݔ‬଴ ‫ݐ‬ =
ଵ
ଶ
‫ݔ‬ ‫ݐ‬ − ‫ݔ‬ −‫ݐ‬
=
ଵ
ଶ
݁ିଶ௧
cos ‫ݐ‬ − ݁ଶ௧
cos(‫)ݐ‬
= cos ‫ݐ‬
௘షమ೟ି௘మ೟
ଶ
= −sinh (2‫)ݐ‬ cos ‫ݐ‬
Prof: Sarun Soman, MIT, Manipal

19. Even-Odd decomposition of a signal
Complex valued signal?
Conjugate Symmetry
A complex valued signal x(t) is said to be conjugate
symmetric if
‫ݔ‬ −‫ݐ‬ = ‫ݔ‬∗(‫)ݐ‬ (1)
Let
‫ݔ‬ ‫ݐ‬ = ܽ ‫ݐ‬ + ݆ܾ(‫)ݐ‬ (2)
‫ݔ‬∗ ‫ݐ‬ = ܽ ‫ݐ‬ − ݆ܾ ‫ݐ‬ (3)
Prof: Sarun Soman, MIT, Manipal

20. Even-Odd decomposition of a signal
Substitute (2) and (3) in (1)
ܽ −‫ݐ‬ + ݆ܾ −‫ݐ‬ = ܽ ‫ݐ‬ − ݆ܾ(‫)ݐ‬
Equating real part
ܽ −‫ݐ‬ = ܽ ‫ݐ‬
Equation imaginary part
ܾ −‫ݐ‬ = −ܾ(‫)ݐ‬
Inference:
A complex valued signal x(t) is conjugate symmetric
if its real part is even and its imaginary part is odd.
Prof: Sarun Soman, MIT, Manipal

21. Example
The signals ‫ݔ‬ଵ ‫ݐ‬ and ‫ݔ‬ଶ(‫)ݐ‬ shown in Figs constitute the
real and imaginary parts, respectively, of a complex
valued signal x(t). What form of symmetry does x(t) have?
Ans:
Conjugate symmetry
Prof: Sarun Soman, MIT, Manipal

22. Periodic and non-periodic signals
A CT periodic signal is a function of time that satisfies the
condition
‫ݔ‬ ‫ݐ‬ = ‫ݔ‬ ‫ݐ‬ + ܶ ݂‫ݐ ݈݈ܽ ݎ݋‬
Where T is a positive constant
Angular frequency ߱ = 2ߨ݂ rad/sec
PERIODIC SIGNAL NON PERIODIC
Prof: Sarun Soman, MIT, Manipal

23. Periodic and non-periodic signals
A DT periodic signal is a function of time that satisfies the
condition
‫ݔ‬ ݊ = ‫ݔ‬ ݊ + ܰ ݂‫݊ ݎ݁݃݁ݐ݊݅ ݎ݋‬
Where N is a +ve integer
Fundamental angular frequency Ω =
ଶగ
ே
rad
PERIODIC
NON PERIODIC
Prof: Sarun Soman, MIT, Manipal

24. Example1
Find the fundamental frequency of the DT square wave shown in
Fig.
Ans:
Ω =
ଶగ
଼
=
గ
ସ
‫ ݀ܽݎ‬
A DT signal can be periodic iff
Ω
ଶగ
is a rational number
Prof: Sarun Soman, MIT, Manipal

25. Example2
Determine if the sinusoidal DT sequence is periodic.
‫ݔ‬ ݊ = cos(0.5݊ + ߠ)
Ans:
‫ݔ‬ ݊ = cos(Ω݊ + ߠ)
Ω
ଶగ
=
଴.ହ
ଶగ
=
ଵ
ସగ
Aperiodic signal
irrationaI
Prof: Sarun Soman, MIT, Manipal

26. Linear combination of two CT signals
Let ݃ ‫ݐ‬ = ‫ݔ‬ଵ ‫ݐ‬ + ‫ݔ‬ଶ(‫)ݐ‬
‫ݔ‬ଵ ‫ݐ‬ is periodic with fundamental period ܶଵ
‫ݔ‬ଶ(‫)ݐ‬is periodic with fundamental period ܶଶ
g(t) periodic?
g(t) will be periodic iff
்భ
்మ
is rational.
The fundamental period of g(t)
݊ܶଵ = ݉ܶଶ
Prof: Sarun Soman, MIT, Manipal

27. Example
Determine if the g(t) is periodic. If yes find the fundamental
period.
݃ ‫ݐ‬ = 3 sin(4 ߨ‫)ݐ‬ + 7 cos(3ߨ‫)ݐ‬
Ans:
ܶଵ =
ଶగ
ఠ
=
ଶగ
ସగ
=
ଵ
ଶ
‫ܿ݁ݏ‬
ܶଶ =
ଶగ
ଷగ
=
ଶ
ଷ
‫ܿ݁ݏ‬
்భ
்మ
=
௠
௡
=
ଷ
ସ
்భ
்మ
is rational –periodic
Fundamental period
݊ܶଵ‫ܶ݉ ݎ݋‬ଶ
2sec
Prof: Sarun Soman, MIT, Manipal

28. Deterministic and Random signals
A deterministic signal is a signal about which there is no
uncertainty with respect to its value at any time.
A random signal is a signal about which there is
uncertainty before it occurs.
EEG signal
Prof: Sarun Soman, MIT, Manipal

29. Energy signal and power signals
Instantaneous power
‫݌‬ ‫ݐ‬ =
௩మ(௧)
ோ
‫݌ ݎ݋‬ ‫ݐ‬ = ܴ݅ଶ
(‫)ݐ‬
If R=1Ω and x(t) represents current or voltage then instantaneous
power is
‫݌‬ ‫ݐ‬ = ‫ݔ‬ଶ
(‫)ݐ‬
Average power?
Average w.r.t to time- time averaged power
Total energy of a CT signal x(t)
‫ܧ‬ = lim
்→ஶ
‫׬‬ ‫ݔ‬ଶ
‫ݐ‬ ݀‫ݐ‬
೅
మ
ି
೅
మ
= ‫׬‬ ‫ݔ‬ଶ
‫ݐ‬ ݀‫ݐ‬
ஶ
ିஶ
Prof: Sarun Soman, MIT, Manipal

30. Energy signal and power signals
Time averaged power or average power
ܲ = lim
்→ஶ
ଵ
்
‫׬‬ ‫ݔ‬ଶ
‫ݐ‬ ݀‫ݐ‬
೅
మ
ି
೅
మ
If the signal is periodic with period T
ܲ =
ଵ
்
‫׬‬ ‫ݔ‬ଶ ‫ݐ‬ ݀‫ݐ‬
೅
మ
ି
೅
మ
DT signal
Total energy of a DT signal
‫ܧ‬ = ∑ ‫ݔ‬ଶ[݊]ஶ
ିஶ
Prof: Sarun Soman, MIT, Manipal

31. Energy signal and power signals
Average power
ܲ = lim
ே→ஶ
ଵ
ଶே
∑ ‫ݔ‬ଶ
[݊]ே
ିே
Average power in a periodic signal x[n] with fundamental
period N
ܲ =
ଵ
ଶே
∑ ‫ݔ‬ଶ
[݊]ே
ିே
Energy signal iff 0 < ‫ܧ‬ < ∞
Power signal iff 0 < ܲ < ∞
Usually Periodic Signals are viewed as Power Signals and Non
Periodic Signals as Energy Signals
Power signal has infinite energy
Energy signal has zero average power
Prof: Sarun Soman, MIT, Manipal

32. Example1
Find the energy of the signal
‫ݔ‬ ݊ = ൝
݊, 0 ≤ ݊ < 5
10 − ݊, 5 ≤ ݊ ≤ 10
0, ‫ݓݏ݅ݓݎ݄݁ݐ݋‬
Ans:
‫ܧ‬ = ∑ ‫ݔ‬ଶ[݊]ஶ
ିஶ
= ∑ ݊ଶ + ∑ (10 − ݊)ଶଵ଴
௡ୀହ
ସ
௡ୀ଴
= 0 + 1 + 4 + 9 + 16 +
[5ଶ + 4ଶ + 3ଶ + 2ଶ + 1ଶ]
= 85‫ܬ‬
Prof: Sarun Soman, MIT, Manipal

33. Example2
Determine if x[n] is power or energy signal.
‫ݔ‬ ݊ = ൜
sin ߨ݊, −4 ≤ ݊ ≤ 4
0, ‫݁ݏ݅ݓݎ݄݁ݐ݋‬
Ans:
‫ܧ‬ = ∑ ‫ݔ‬ଶ[݊]ஶ
ିஶ
= ∑ ‫݊݅ݏ‬ଶߨ݊ସ
ିସ
= ∑
ଵିୡ୭ୱ ଶగ௡
ଶ
ସ
ିସ
= 0
P=0 aperiodic
Neither energy nor power signal
Prof: Sarun Soman, MIT, Manipal

34. Example3
‫ݔ‬ ‫ݐ‬ = ൝
‫,ݐ‬ 0 ≤ ‫ݐ‬ ≤ 1
2 − ‫,ݐ‬ 1 ≤ ‫ݐ‬ ≤ 2
0, ‫݁ݏ݅ݓݎ݄݁ݐ݋‬
Energy or power signal?
Ans:
Aperiodic
‫ܧ‬ = ‫׬‬ ‫ݔ‬ଶ
‫ݐ‬ ݀‫ݐ‬
ஶ
ିஶ
= ‫׬‬ ‫ݐ‬ଶ݀‫ݐ‬ + ‫׬‬ (2 − ‫)ݐ‬ଶ݀‫ݐ‬
ଶ
ଵ
ଵ
଴
=
ଵ
ଷ
+ ‫׬‬ (4 − 4‫ݐ‬ + ‫ݐ‬ଶ) ݀‫ݐ‬
ଶ
ଵ
=
ଶ
ଷ
‫ܬ‬
Finite energy –Energy signal
Prof: Sarun Soman, MIT, Manipal

35. Tutorial
Find the even and odd components of each of
the following signals.
a) ‫ݔ‬ ‫ݐ‬ = cos ‫ݐ‬ + sin ‫ݐ‬ + sin ‫ݐ‬ cos ‫ݐ‬
b) ‫ݔ‬ ‫ݐ‬ = 1 + ‫ݐ‬ + 3‫ݐ‬ଶ
+ 5‫ݐ‬ଷ
+ 9‫ݐ‬ସ
c) ‫ݔ‬ ‫ݐ‬ = 1 + ‫ݐ‬ cos ‫ݐ‬ + ‫ݐ‬ଶ
sin ‫ݐ‬ + ‫ݐ‬ଶ
sin ‫ݐ‬ cos ‫ݐ‬
d) ‫ݔ‬ ‫ݐ‬ = (1 + ‫ݐ‬ଶ
)ܿ‫ݏ݋‬ଷ
(10‫)ݐ‬
Prof: Sarun Soman, MIT, Manipal

36. Tutorial
a) ‫ݔ‬ ‫ݐ‬ = cos ‫ݐ‬ + sin ‫ݐ‬ + sin ‫ݐ‬ cos ‫ݐ‬
= cos ‫ݐ‬ + sin ‫ݐ‬ 1 + cos ‫ݐ‬
‫ݔ‬ −‫ݐ‬ = cos ‫ݐ‬ − sin ‫ݐ‬ (1 + cos ‫)ݐ‬
‫ݔ‬௘ ‫ݐ‬ = cos ‫ݐ‬
‫ݔ‬଴ ‫ݐ‬ = sin ‫ݐ‬ (1 + cos ‫)ݐ‬
b) ‫ݔ‬ ‫ݐ‬ = 1 + ‫ݐ‬ + 3‫ݐ‬ଶ + 5‫ݐ‬ଷ + 9‫ݐ‬ସ
‫ݔ‬ −‫ݐ‬ = 1 − ‫ݐ‬ + 3‫ݐ‬ଶ − 5‫ݐ‬ଷ + 9‫ݐ‬ସ
‫ݔ‬௘ ‫ݐ‬ = 1 + 3‫ݐ‬ଶ + 9‫ݐ‬ସ
‫ݔ‬଴ ‫ݐ‬ = ‫ݐ‬ + 5‫ݐ‬ଷ
Prof: Sarun Soman, MIT, Manipal

37. Tutorial
c) ‫ݔ‬ ‫ݐ‬ = 1 + ‫ݐ‬ cos ‫ݐ‬ + ‫ݐ‬ଶ sin ‫ݐ‬ + ‫ݐ‬ଷ sin ‫ݐ‬ cos ‫ݐ‬
‫ݔ‬ −‫ݐ‬ = 1 − ‫ݐ‬ cos ‫ݐ‬ − ‫ݐ‬ଶ
sin ‫ݐ‬ + ‫ݐ‬ଷ
sin ‫ݐ‬ cos ‫ݐ‬
‫ݔ‬௘ ‫ݐ‬ = 1 + ‫ݐ‬ଷ
sin ‫ݐ‬ cos ‫ݐ‬
‫ݔ‬଴ ‫ݐ‬ = ‫ݐ‬ cos ‫ݐ‬ + ‫ݐ‬ଶ sin ‫ݐ‬
d) ‫ݔ‬ ‫ݐ‬ = (1 + ‫ݐ‬ଷ)ܿ‫ݏ݋‬ଷ(10‫)ݐ‬
= ܿ‫ݏ݋‬ଷ 10‫ݐ‬ + ‫ݐ‬ଷܿ‫ݏ݋‬ଷ(10‫)ݐ‬
‫ݔ‬ −‫ݐ‬ = ܿ‫ݏ݋‬ଷ 10‫ݐ‬ − ‫ݐ‬ଷܿ‫ݏ݋‬ଷ(10‫)ݐ‬
‫ݔ‬௘ ‫ݐ‬ = ܿ‫ݏ݋‬ଷ 10‫ݐ‬
‫ݔ‬଴ ‫ݐ‬ = ‫ݐ‬ଷܿ‫ݏ݋‬ଷ(10‫)ݐ‬
Prof: Sarun Soman, MIT, Manipal

38. Tutorial
For each of the following signals, determine whether it is
periodic, and if it is, find the fundamental period.
a) ‫ݔ‬ ‫ݐ‬ = ܿ‫ݏ݋‬ଶ 2ߨ‫ݐ‬
b) ‫ݔ‬ ‫ݐ‬ = ‫݊݅ݏ‬ଷ 2‫ݐ‬
c) ‫ݔ‬ ‫ݐ‬ = ݁ିଶ௧ cos(2ߨ‫)ݐ‬
a) ‫ݔ‬ ‫ݐ‬ = ܿ‫ݏ݋‬ଶ 2ߨ‫ݐ‬
‫ݔ‬ ‫ݐ‬ =
ଵାୡ୭ୱ ସగ௧
ଶ
= 0.5 + 0.5 cos 4ߨ‫ݐ‬ periodic
ܶ =
ଶగ
ఠ
=
ଶగ
ସగ
=
ଵ
ଶ
‫ݏ‬
Prof: Sarun Soman, MIT, Manipal

39. Tutorial
b) ‫ݔ‬ ‫ݐ‬ = ‫݊݅ݏ‬ଷ
2‫ݐ‬
‫݊݅ݏ‬ଷ
‫ݔ‬ =
ଷ ୱ୧୬ ௫ିୱ୧୬ ଷ௫
ସ
‫ݔ‬ ‫ݐ‬ =
ଷ ୱ୧୬ ଶ௧ିୱ୧୬ ଺௧
ସ
ܶଵ =
ଶగ
ଶ
= ߨ‫ݏ‬
ܶଶ =
ଶగ
଺
=
గ
ଷ
‫ݏ‬
்భ
்మ
=
௠
௡
=
ଷ
ଵ
rational- x(t)periodic
ܶ = ߨ‫ݏ‬
Prof: Sarun Soman, MIT, Manipal

40. Tutorial
c) ‫ݔ‬ ‫ݐ‬ = ݁ିଶ௧
cos 2ߨ‫ݐ‬
Non-periodic
݁ିଶ௧
Prof: Sarun Soman, MIT, Manipal

41. Tutorial
‫ݔ‬ ݊ = (−1)௡
Periodic with N=2 samples
n x[n]
0 1
1 -1
2 1
3 -1
Prof: Sarun Soman, MIT, Manipal

42. Tutorial
1. ‫ݔ‬ ݊ = (−1)௡మ
Periodic with N=2 samples
2. ‫ݔ‬ ݊ = cos 2݊
Ω
ଶగ
=
ଵ
గ
irrational –non periodic
n ࢔૛ x[n]
0 0 1
1 1 -1
2 4 1
3 9 -1
4 16 1
Prof: Sarun Soman, MIT, Manipal

43. Tutorial
Categorize each of the following signals as an energy signal or a power
signal, and find the energy or time-averaged power.
1.‫ݔ‬ ‫ݐ‬ = 5 cos ߨ‫ݐ‬ + sin 5ߨ‫ݐ‬ , −∞ < ‫ݐ‬ < ∞
Periodic or aperiodic?
ܶଵ = 2 and ܶଶ =
ଶ
ହ
்భ
்మ
= 5 periodic
Fundamental period T=2
ܲ =
ଵ
்
‫׬‬ ‫ݔ‬ଶ
‫ݐ‬ ݀‫ݐ‬
೅
మ
ି
೅
మ
=
ଵ
ଶ
‫׬‬ (25ܿ‫ݏ݋‬ଶ
ߨ‫ݐ‬ + 10 cos ߨ‫ݐ‬ sin 5ߨ‫ݐ‬ + ‫݊݅ݏ‬ଶ
5ߨ‫)ݐ‬
ଵ
ିଵ
݀‫ݐ‬
Prof: Sarun Soman, MIT, Manipal

44. Tutorial
=
ଶହ
ସ
‫׬‬ (1 + cos 2ߨ‫)ݐ‬
ଵ
ିଵ
݀‫ݐ‬ + 5 ‫׬‬ cos ߨ‫ݐ‬ sin 5ߨ‫ݐ‬
ଵ
ିଵ
݀‫ݐ‬
+
ଵ
ସ
‫׬‬ 1 − cos 10ߨ‫ݐ‬ ݀‫ݐ‬
ଵ
ିଵ
=
ଶହ
ଶ
+
ଵ
ଶ
+
ହ
ଶ
‫׬‬ sin 4ߨ‫ݐ‬ + sin 6ߨ‫ݐ‬ ݀‫ݐ‬
ଵ
ିଵ
=
ଶହ
ଶ
+
ଵ
ଶ
= 13ܹ
Prof: Sarun Soman, MIT, Manipal

45. Tutorial
Classify the signals as energy or power signals
-2 20
5
0-2-4-6 2 4 6
5
(a)
(b)
Prof: Sarun Soman, MIT, Manipal

46. Tutorial
a)Aperiodic
‫ܧ‬ = ‫׬‬ 5ଶ݀‫ݐ‬
ଶ
ିଶ
= 25 ∗ 4 = 100‫ܬ‬
Finite energy – energy signal
b)Periodic- T=8s
ܲ =
ଵ
்
‫׬‬ ‫ݔ‬ଶ ‫ݐ‬ ݀‫ݐ‬
೅
మ
ି
೅
మ
=
ଵ
଼
‫׬‬ 5ଶ
݀‫ݐ‬
ଶ
ିଶ
=
ଵ଴଴
଼
= 12.5ܹ
Finite power- power signal
Prof: Sarun Soman, MIT, Manipal

47. Basic operation on signals
Operations on dependent variables
1. Amplitude scaling
2. Addition
3. Multiplication
4. Differentiation
5. Integration
Operations in independent variables
1. Time scaling
2. Time shifting
Prof: Sarun Soman, MIT, Manipal

48. Operations on dependent variables
Amplitude scaling
‫ݕ‬ ‫ݐ‬ = ܿ‫)ݐ(ݔ‬
‫ݕ‬ ݊ = ܿ‫]݊[ݔ‬
C is the scaling factor.
Eg. Amplifier
Addition
‫ݕ‬ଵ ‫ݐ‬ = ‫ݔ‬ଵ(‫)ݐ‬ + ‫ݔ‬ଶ(‫)ݐ‬
‫ݕ‬ ݊ = ‫ݔ‬ଵ ݊ + ‫ݔ‬ଶ ݊
Eg. Audio mixer
Multiplication
‫ݕ‬ଵ ‫ݐ‬ = ‫ݔ‬ଵ(‫)ݐ‬ ‫ݔ‬ଶ(‫)ݐ‬
‫ݕ‬ ݊ = ‫ݔ‬ଵ ݊ ‫ݔ‬ଶ ݊
Eg. AM radio signal
Prof: Sarun Soman, MIT, Manipal

49. Operations on dependent variables
Differentiation
‫ݕ‬ ‫ݐ‬ =
ௗ
ௗ௧
‫)ݐ(ݔ‬
Eg.
‫ݒ‬ ‫ݐ‬ = ‫ܮ‬
݀
݀‫ݐ‬
݅(‫)ݐ‬
Integration
‫ݕ‬ ‫ݐ‬ = ‫׬‬ ‫)߬(ݔ‬
௧
ିஶ
݀߬
Prof: Sarun Soman, MIT, Manipal

50. Operations on dependent variables
Eg.
‫ݒ‬ ‫ݐ‬ =
1
‫ܥ‬
න ݅ ߬ ݀߬
௧
ିஶ
Prof: Sarun Soman, MIT, Manipal

51. Operations on independent variables
Time scaling
‫ݕ‬ ‫ݐ‬ = ‫)ݐܽ(ݔ‬
If a>1 – signal compression
If 0<a<1- signal expansion
‫ݕ‬ ݊ = ‫ݔ‬ ݇݊ , ݇ > 0
k can have only integer values
k>1 –signal compression
Prof: Sarun Soman, MIT, Manipal

52. Operations on independent variables
Compression of DT signal results in some
loss of information
Prof: Sarun Soman, MIT, Manipal

53. Operations on independent variables
Reflection
‫ݕ‬ ‫ݐ‬ = ‫ݔ‬ −‫ݐ‬
‫)ݐ(ݕ‬ represents a reflected version of ‫)ݐ(ݔ‬about t=0
‫ݕ‬ ݊ = ‫]݊−[ݔ‬
Prof: Sarun Soman, MIT, Manipal

54. Operations on independent variables
Time shifting
‫ݕ‬ ‫ݐ‬ = ‫ݐ(ݔ‬ − ‫ݐ‬଴)
‫ݐ‬଴ > 0 right shift and ‫ݐ‬଴ < 0 left shift
Figure shows a rectangular pulse x(t) of unit amplitude and unit
duration. Find ‫ݕ‬ ‫ݐ‬ = ‫ݐ(ݔ‬ − 2).
Prof: Sarun Soman, MIT, Manipal

55. Operations on independent variables
In the case of DT signal
‫ݕ‬ ݊ = ‫݊[ݔ‬ − ݉], m is an integer
The DT signal ‫ݔ‬ ݊ = ൝
1, ݊ = 1,2
−1, ݊ = −1, −2
0, ݊ = 0ܽ݊݀ ݊ > 2
.Find the time-shifted
signal ‫ݕ‬ ݊ = ‫ݔ‬ ݊ + 3 .
Prof: Sarun Soman, MIT, Manipal

56. Operations on independent variables
n x[n] y[n]=x[n+3]
-5 0 y[-5]=x[-2]
-4 0 y[-4]=x[-1]
-3 0 y[-3]=x[0]
-2 -1 y[-2]=x[1]
-1 -1 y[-1]=x[2]
0 0 y[0]=x[3]
1 1 y[1]=x[4]
2 1 y[2]=x[5]
●-2-4
●●● ●
2 4
x[n]
n
●● ● ●
-4
-2 2
y[n]
n
4
●●
Prof: Sarun Soman, MIT, Manipal

57. Operations on independent variables
Precedence Rule For Time Shifting And Time Scaling
Let ‫ݕ‬ ‫ݐ‬ = ‫ݐܽ(ݔ‬ − ܾ)
To obtain y(t) from x(t) time-shifting and time scaling
must be performed in the correct order.
1.Perform time-shifting first
‫ݒ‬ ‫ݐ‬ = ‫ݐ(ݔ‬ − ܾ) – intermediate signal
2.Perform Scaling
‫ݕ‬ ‫ݐ‬ = ‫)ݐܽ(ݒ‬
= ‫ݐܽ(ݔ‬ − ܾ)
Prof: Sarun Soman, MIT, Manipal

58. Example1
Consider the rectangular pulse x(t) of unit amplitude and a
duration of 2 time units, depicted in Fig.Find ‫ݕ‬ ‫ݐ‬ = ‫ݐ2(ݔ‬ + 3).
Ans: b=-3, a=2
Prof: Sarun Soman, MIT, Manipal

59. Example2
A DT signal is defined by ‫ݔ‬ ݊ = ൝
1, ݊ = 1,2
−1, ݊ = −1, −2
0, ݊ = 0ܽ݊݀ ݊ > 2
.
Find ‫ݕ‬ ݊ = ‫ݔ‬ 2݊ + 3 .
Ans: ‫ݕ‬ ݊ = ‫݊݇[ݔ‬ − ݉]
k=2,m=-3
Prof: Sarun Soman, MIT, Manipal

60. Elementary Signals
Exponential Signals
1. Exponential Signals
2. Sinusoidal Signals
3. Step function
4. Ramp function
Serve as building blocks for the construction of
more complex signals.
Can be used to model many physical signals that
occur in nature.
Prof: Sarun Soman, MIT, Manipal

61. Elementary Signals
Exponential Signal
‫ݔ‬ ‫ݐ‬ = ‫݁ܤ‬ି௔௧
a<0 decaying exponential
a>0 growing exponential
‫ݔ‬ ݊ = ‫ݎܤ‬௡
r>1 growing exponential
r<1 decaying exponential
Complex exponential signals
݁௝⍵௧ and ݁௝Ω௡
Prof: Sarun Soman, MIT, Manipal

62. Elementary Signals
Sinusoidal Signals
‫ݔ‬ ‫ݐ‬ = ‫ܣ‬ cos(⍵‫ݐ‬ + ⏀)
ܶ =
ଶగ
ఠ
Proof:
‫ݔ‬ ‫ݐ‬ + ܶ = ‫ܣ‬ cos(⍵(‫ݐ‬ + ܶ) + ⏀)
= ‫ܣ‬ cos(⍵‫ݐ‬ + ⍵ܶ + ⏀)
= ‫ܣ‬ cos(⍵‫ݐ‬ + 2ߨ + ⏀)
= ‫ܣ‬ cos(⍵‫ݐ‬ + ⏀)
= ‫)ݐ(ݔ‬
Prof: Sarun Soman, MIT, Manipal

63. Elementary Signals
DT sinusoid
‫ݔ‬ ݊ = ‫ܣ‬ cos(Ω݊ + ∅)
Proof:
‫ݔ‬ ݊ + ܰ = ‫ܣ‬ cos(Ω(݊ + ܰ) + ∅)
= ‫ܣ‬ cos(Ω݊ + Ωܰ + ∅)
‫ݔ‬ ݊ = ‫ݔ‬ ݊ + ܰ iff Ωܰ is an integer multiple of 2ߨ
Ωܰ = 2ߨ݉
Ω
2ߨ
=
݉
ܰ
Prof: Sarun Soman, MIT, Manipal

64. Elementary Signals
Exponentially Damped Sinusoidal Signals
‫ݔ‬ ‫ݐ‬ = ‫݁ܣ‬ିఈ௧ sin(߱‫ݐ‬ + ∅), ߙ > 0
Step function
The DT version of the unit step function is defined by
‫ݑ‬ ݊ = ൜
1, ݊ ≥ 0
0, ݊ < 0
The CT time version of the unit step function is defined by
‫ݑ‬ ‫ݐ‬ = ൜
1, ‫ݐ‬ > 0
0, ‫ݐ‬ < 0
1
●●●
0 2 4-1
u[n]
n
Prof: Sarun Soman, MIT, Manipal

65. Example 1
Consider the rectangular pulse x(t) shown in Fig. The pulse has
an amplitude A and duration of 1s. Express x(t) as a weighted
sum of two step function.
Ans: ࡭࢛(࢚ + ૙. ૞)
࡭࢛(࢚ − ૙. ૞)
࡭࢛(࢚ + ૙. ૞) − ࡭࢛(࢚ − ૙. ૞)
‫ݔ‬ ‫ݐ‬ = ‫ݐ(ݑܣ‬ + 0.5) − ‫ݐ(ݑܣ‬ − 0.5)
Prof: Sarun Soman, MIT, Manipal

66. Example 2
A DT signal ‫ݔ‬ ݊ = ൜
1, 0 ≤ ݊ ≤ 9
0, ‫݁ݏ݅ݓݎ݄݁ݐ݋‬
.Using u[n], describe x[n] as
superposition of two step functions.
Ans:
1
u[n]
●●●
0 2 4-1
n6 8 10
0
●●●
10 12 148
n16 18
● ● ● ● ● ●●
u[n-10]
‫ݔ‬ ݊ = ‫ݑ‬ ݊ − ‫݊[ݑ‬ − 10]
Prof: Sarun Soman, MIT, Manipal

67. Elementary Signals
Ramp function
‫ݎ‬ ‫ݐ‬ = ൜
‫,ݐ‬ ‫ݐ‬ ≥ 0
0, ‫ݐ‬ < 0
or
‫ݎ‬ ‫ݐ‬ = ‫]ݐ[ݑݐ‬
Note: Integral of unit step function u(t) is a ramp function of unit
slope.
‫ݎ‬ ݊ = ൜
݊, ݊ ≥ 0
0, ݊ < 0
or
‫ݎ‬ ݊ = ݊‫]݊[ݑ‬ ●●●● n
r[n]
0
Prof: Sarun Soman, MIT, Manipal

68. Elementary Signals
Impulse function
DT version of the unit impulse if defined by
‫ݔ‬ ݊ = ൜
1, ݊ = 0
0, ݊ ≠ 0
CT version of unit impulse
ߜ ‫ݐ‬ = 0 ݂‫ݐ ݎ݋‬ ≠ 0 (1)
and
‫׬‬ ߜ ‫ݐ‬ ݀‫ݐ‬ = 1
ஶ
ିஶ
(2)
(1) says that impulse is zero every where except at
origin.
(2) Says total area under the unit impulse is unity.
Prof: Sarun Soman, MIT, Manipal

69. Elementary Signals
Evolution of a rectangular pulse of unit area into an impulse of
unit strength.
As duration decreases , the rectangular pulse approximates the
impulse more closely.
Mathematical relation between impulse and rectangular pulse
function.
ߜ ‫ݐ‬ = lim
∆→଴
‫)ݐ(ݔ‬
∆ ∆
∆
Prof: Sarun Soman, MIT, Manipal

70. Elementary Signals
ߜ(‫)ݐ‬ is the derivative of ‫ݑ‬ ‫ݐ‬
Sifting property
‫׬‬ ‫ݐ(ߜ)ݐ(ݔ‬ − ‫ݐ‬଴)
ஶ
ିஶ
݀‫ݐ‬ = ‫ݔ‬ ‫ݐ‬଴
Sifts out the value of ‫)ݐ(ݔ‬ at time ‫ݐ‬ = ‫ݐ‬଴
‫)ݐ(ݔ‬
‫ݐ(ݔ‬଴)
ߜ(‫ݐ‬ − ‫ݐ‬଴)
‫ݐ‬଴
Prof: Sarun Soman, MIT, Manipal

71. Elementary Signals
Time scaling property
ߜ ܽ‫ݐ‬ =
ଵ
௔
ߜ ‫ݐ‬ , ܽ > 0
Proof:
ߜ ‫ݐ‬ = lim
∆→଴
‫)ݐ(ݔ‬
Replacing ‘t’ with ‘at’
ߜ ܽ‫ݐ‬ = lim
∆→଴
‫)ݐܽ(ݔ‬ (1)
Prof: Sarun Soman, MIT, Manipal

72. Elementary Signals
(1) becomes
lim
∆→଴
ܽ‫)ݐܽ(ݔ‬ = ߜ(‫)ݐ‬
lim
∆→଴
‫)ݐܽ(ݔ‬ =
ଵ
௔
ߜ(‫)ݐ‬
ߜ ܽ‫ݐ‬ =
ଵ
௔
ߜ(‫)ݐ‬
∆
∆
∆
∆
∆
∆
Prof: Sarun Soman, MIT, Manipal

73. Tutorials
A triangular pulse signal x(t) is depicted in Fig. Sketch each of the
following signals derived from x(t).
a) x(3t)
b) x(3t+2)
c) x(-2t-1)
d) x(2(t+2))
e) x(2(t-2))
f) x(3t)+x(3t+2)
Prof: Sarun Soman, MIT, Manipal

74. Tutorials
a)x(3t) c) x(-2t-1)
b)
-1-3 -2 0
‫ݐ(ݔ‬ + 2)
t
-1
−
1
3
0
‫ݐ3(ݔ‬ + 2)
t
1 20
‫ݐ(ݔ‬ − 1)
t
10
‫ݐ2(ݔ‬ − 1)
t
0-1
‫ݐ2−(ݔ‬ − 1)
t
Prof: Sarun Soman, MIT, Manipal

75. Tutorial
e) f)
d)
Prof: Sarun Soman, MIT, Manipal

76. Tutorial
Sketch ‫ݔ‬ ݊ ‫3[ݑ‬ − ݊]
1
●●●
0 2 4-1
u[n]
n
_ _ _ _
1
●●●
-2 0 2-4
u[n+3]
n4
_ _ _ _
●● ● ●0 2 4 6-2-4
x[n]
n
2
●
●●● ● ●0 2 4 6-2-4
x[n]u[-n+3]
n
2
●
1
● ● ●
-2 0 2-4
u[-n+3]
n
_ _ _ _
4
●● ● ●0 2 4 6-2-4
x[n]
n
2
●
Prof: Sarun Soman, MIT, Manipal

77. Tutorials
Given x[n] & y[n] Find
1) ‫݊3[ݔ‬ − 1]
2) ‫ݕ‬ 2 − 2݊
3) ‫ݔ‬ 2݊ + ‫݊[ݕ‬ − 4]1
● ●●
-2 0 2-4
x[n]
n
4
2
3
1
●●● -2
0 2
-4
y[n]
n4
-6
● ●
6
Prof: Sarun Soman, MIT, Manipal

78. Tutorial
1. ‫ݔ‬ 3݊ − 1 2. ‫ݕ‬ −2݊ + 2
● ●●
-2 0 2-4
v[n]=x[n-1]
n
4●
● ● ●
-2 0 2-4
v[3n]=x[3n-1]
n
4
● ● ●
1
● ●
-2
0 2
-4
v[n]= y[n+2]
n
4
●-6
1
●● -2 0
2
-4
v[-n]
n4● 6
●
1
● ●
-2 0
2
-4
v[-2n]
n
4
●
6
● ●
Prof: Sarun Soman, MIT, Manipal

79. Tutorial
3. ‫ݔ‬ 2݊ + ‫ݕ‬ ݊ − 4
● ●●
-2 0 2
x[2n]
n
2
1
●●●
-2 0 2
8
y[n-4]
n
4
● ●6 10
1
●●
-2 0 2
8
x[2n]+y[n-4]
n
4
● ●6
Prof: Sarun Soman, MIT, Manipal

80. Tutorial
Find the odd and even part of
signal x(t).
Even
Odd
-1 0 1 2 t
x(t)
-1
1
-1 0 1-2 t
x(-t)
-1
1
0 2 t
0.5[x(t)+x(-t)]
-0.5
1
-2
0
2
t
0.5[x(t)-x(-t)]
-0.5
0.5
-2
Prof: Sarun Soman, MIT, Manipal

81. Tutorial
A CT signal x(t) is shown. Find
‫ݕ‬ ‫ݐ‬ = ‫ݔ‬ ‫ݐ‬ ߜ ‫ݐ‬ +
ଷ
ଶ
− ߜ ‫ݐ‬ −
ଷ
ଶ
Sifting property
න ‫ݐ(ߜ)ݐ(ݔ‬ − ‫ݐ‬଴)
ஶ
ିஶ
݀‫ݐ‬ = ‫ݔ‬ ‫ݐ‬଴
-1 0 1 2 3 t
x(t)
1
2
ߜ ‫ݐ‬ +
3
2 ߜ ‫ݐ‬ −
3
2
0
2ߜ ‫ݐ‬ −
ଷ
ଶ
0
Prof: Sarun Soman, MIT, Manipal

82. Tutorial
Sketch the following signals.
1. ‫ݔ‬ ݊ =
ଵ
ଶ
, 1, 2, 3, 4, 8
0 2
1
3
Prof: Sarun Soman, MIT, Manipal

83. Tutorial
x(t) and y(t) are shown in Fig.
Sketch
1. ‫ݔ‬ ‫ݐ‬ − 1 ‫)ݐ−(ݕ‬
2. ‫ݔ‬ ‫ݐ‬ ‫ݐ−(ݕ‬ − 1)
43210
-1
1
x(t-1)
t
t
-1
-2 -1 0 1 2
y(-t)
1
t
-1
-2 -1 0 1 2
y(t)
1
t
x(t-1)y(-t)
1
0 1 2
3210
-1
-1
1
x(t)
t
Prof: Sarun Soman, MIT, Manipal

84. Tutorial
‫ݔ‬ ‫ݐ‬ ‫ݐ−(ݕ‬ − 1)
t
-1
-1 0 1-2 2
y(t-1)
1
t
-1
-1 0 1-2-3
y(-t-1)
1
0-1 1
1
x(t)y(-t-1)
3210
-1
-1
1
x(t)
t
3
Prof: Sarun Soman, MIT, Manipal