Signal and Systems
B. TECH./MBA TECH. – EXTC/IT
Sem III
AY 2023-2024
Instructor: Dr. Prashant Kharote
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What is a Signal?
• Conveys information
• Sound
• Image
• Video
• Biomedical data
• Music
• Speech
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How does a Signal look ?
Communication
signal
Electrical
signal
Speech /
Audio
signal
Biomedical
signal
Images
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Signal Representation
Signal is a function of an independent variable
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Signal
Modify signal in time
domain
Analyze
Processing
Unit 3:
Fourier Series
Unit 1:
Elementary signals and
operations on signals
(frequency analysis of periodic CT signal)
Unit 4:
Fourier Transform
(frequency analysis Aperiodic CT signal)
Unit 5:
Laplace Transform
(analyze CT signal in complex domain, s)
Unit 6:
SNSZ Transform (Discrete signals)
Unit 2:
LTI Systems
What is a System :
• We need to process the signal for
• Analysis
• Interpretation
• Manipulation
• Control of signals such as Sound , Images, Video, Biomedical data, control signals,
telecommunication transmission signals, sensor data, time-varying
measurements, etc…
Processing of the signal helps to estimate the characteristic parameters of the signal and
also to transform the signal into desired form.
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Scope of the subject
• To build up the concepts required to analyze and transform signals
• To build up the concepts required to analyze , model and design systems
System:
Filter, Amplifier, mixers ,
modulators
Antenna
Unit 1
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Basic Classification of Signals
Continuous Time Signal
Discrete Time Signal
𝑥(𝑡)
𝑥[𝑛]
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Continuous Time and Discrete Time signal
• Continuous Time (CT) signals are functions whose amplitude continuously varies with time,
• They are denoted as x(t)
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Discrete Time (DT) Signals
• Discrete time signals are functions of values taken as discrete instant of time
• It is often derived from CT signals by sampling at uniform time (sampling period)
• DT signals are represented as x[n] where n denotes discrete time
• n = 0, ±1, ±2, ±3, ….. (note no fractional values)
𝐷𝑖𝑠𝑐𝑟𝑒𝑡𝑒 𝑡𝑖𝑚𝑒 𝑠𝑖𝑔𝑛𝑎𝑙𝑠
𝑎𝑟𝑒 𝑟𝑒𝑝𝑟𝑒𝑠𝑒𝑛𝑡𝑒𝑑 𝑎𝑠
𝑥 𝑛 𝑤ℎ𝑒𝑟𝑒, 𝑛 = 0, ±1, ±2, …
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Example of CT and DT signal
𝑅𝑒𝑝𝑟𝑒𝑠𝑒𝑛𝑡𝑎𝑡𝑖𝑜𝑛 𝑖𝑛 𝐶𝑇 𝑑𝑜𝑚𝑎𝑖𝑛
𝑅𝑒𝑝𝑟𝑒𝑠𝑒𝑛𝑡𝑎𝑡𝑖𝑜𝑛 𝑖𝑛 𝐷𝑇 𝑑𝑜𝑚𝑎𝑖𝑛
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Example of DT signal
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Elementary Continuous Time (CT)
signals / functions
(i) Unit step signal, u(t)
(ii) Unit impulse signal, δ(t)
(iii) Ramp signal, r(t)
(iv) Signum function, Sgn(t)
(v) Exponential Signal, e-at
(vi) Rectangular Signal, rect(t)
(vii)Triangular Signal,
(viii)Sinusoidal Signal
(ix) Sinc Function
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(i) Unit step function
• Unit step function is denoted by u(t). It is defined as
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(ii) Unit impulse function
Unit Impulse function also known as Dirac or delta function
is denoted by δ(t). and it is defined as
Impulse signal exists only at t = 0
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Significance of impulse function
❑To convert a continuous time signal to discrete time signal
❑ SAMPLING: to convert CT signal into DT signal
❑By multiplying CT signal by a unit impulse and its shifted versions
Any signal can be represented as a series of scaled time shifted impulses
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(iii) Ramp signal
• Ramp signal is denoted by r(t), and it is defined as
Ramp signal can also be written as
• Unit ramp signal has a slope of 1
r (t) = t u(t)
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(iv) Signum Function
• Signum function is denoted as sgn(t). It is defined as
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(v) Exponential Signal
• Exponential signal is in the form of
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(vi) Triangular signal / function
𝑥 𝑡 = 2 1 − |𝑡|
|𝑡|
𝑥 𝑡 =𝐴 1−
𝑇
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|𝑡|
𝑥 𝑡 =9 1−
3
(vii) Rectangular signal
𝑡
𝑥 𝑡 = 𝐴 𝑟𝑒𝑐𝑡
𝑇
Example:
5
𝑡
𝑝𝑙𝑜𝑡 ∶ 𝑥 𝑡 = 5 𝑟𝑒𝑐𝑡
4
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−4
2
4
2
−𝟐
𝟐
Example of rectangular signal
𝑝𝑙𝑜𝑡 ∶ 𝑥 𝑡 = 4 𝑟𝑒𝑐𝑡 4𝑡
4
−1/4
2
−𝟏/𝟖
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1/4
2
1/𝟖
Example of rectangular signal
2𝑡
𝑝𝑙𝑜𝑡 ∶ 𝑥 𝑡 = 3 𝑟𝑒𝑐𝑡
𝑇
3
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−𝑇/2
2
𝑇/2
2
−𝑻/𝟒
𝑻/𝟒
Unit Rectangular Signal
1
𝑎
-
1
𝑎
+
1
𝑎
2
𝑇=
𝑎
1
𝑎𝑡
𝑥 𝑡 = 𝑟𝑒𝑐𝑡
𝑎
2
𝐴𝑟𝑒𝑎 𝑤𝑖𝑡ℎ 𝑖𝑛SNS𝑡ℎ𝑒 𝑝𝑢𝑙𝑠𝑒 = 1
1
𝑥 𝑡 = ,
𝑎
1
1
− <𝑇<
𝑎
𝑎
(viii) Sinusoidal Signal
Wo is fundamental frequency
Φ is the phase
To is the fundamental period
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(ix) Sinc Function
• It is denoted as sinc(t) and it is defined as
Peak Value at origin = 1 and zero crossings at ± n
This is a normalized function.
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(x) Sampling Function
• It is denoted as sa(t) and it is defined as
Peak value at origin = 1 and zero crossings at ± nπ
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This is a unnormalized
function.
Elementary Discrete Time (DT)
signals / functions
(i) Unit step signal, u[n]
(ii) Unit impulse signal, δ[n]
(iii) Ramp signal, r[n]
(iv) Signum function, Sgn[n]
(v) Exponential Signal, ean
(vi) Rectangular Signal, rect[n]
(vii)Triangular Signal,
(viii)Sinusoidal Signal
(ix) Sinc Function
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Discrete time signals
1. Discrete time unit step signal
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Discrete time signals
2. Discrete time unit impulse signal
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Relation between DT unit step and unit impulse signal
-
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Discrete time signal
• Exponential signal
Where, C is the scaling factor
α is the shaping factor
For simplicity C = 1
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(c) -1 < α < 0, Double sided Decaying exponent
(d) α < -1 Double sided growing exponent
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Unit 1
❑ Signals
1.
2.
3.
4.
Signals
Elementary signals – analog and discrete signals
Basic operation on signals
Classification of Signals
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Basic Operation on Signals
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Basic Signal Operations
• There are two variable parameters in any signal:
I. Amplitude
II. Time
I.
The following operation can be performed with amplitude:
Amplitude Scaling: a.x(t)
Amplitude Reversal : - x(t)
Addition of signals: x1(t) + x2(t)
II. The following operations can be performed with time:
Subtraction of signals : x1(t) - x2(t)
Time Shifting:
x(t – to)
Multiplication of signals : x1(t) . x2(t)
Time Scaling:
x(at)
Convolution of signals : x1(t) * x2(t)
Time Reversal : x ( - t )
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Amplitude Scaling
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Addition of Signals
Subtraction of signals
Y 𝑡 = 𝑋1 𝑡 − 𝑋2 𝑡
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Multiplication
of signals
Y 𝑡 = 𝑋1 𝑡 − 𝑋2 𝑡
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Example 2:
Plot the signal
a. Y(t) = u(t) + 4 u(t)
a. Y(t) = u(t) + 4 u(t)
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Amplitude Reversal
• Whenever the amplitude of a signal is multiplied by -1, then it is known as
amplitude reversal.
• In this case, the signal produces its mirror image about X-axis. Mathematically,
this can be written as:
𝑦 𝑡 = −𝑥(𝑡)
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• Example : plot
a. 𝑥 𝑡 = −3 𝑢 𝑡
b. 𝑥 𝑡 = 3 𝑢 − 𝑡
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Time reversal of a signal
• If X(t) is the original signal
• Time reversal results in signal such that : Y(t) = X(- t)
𝐴𝑚𝑝𝑙𝑖𝑡𝑢𝑑𝑒 𝑟𝑒𝑣𝑒𝑟𝑠𝑎𝑙 ∶ 𝑡𝑎𝑘𝑒 𝑚𝑖𝑟𝑟𝑜𝑟
𝑖𝑚𝑎𝑔𝑒 𝑎𝑏𝑜𝑢𝑡 𝑋 𝑎𝑥𝑖𝑠
𝑇𝑖𝑚𝑒 𝑟𝑒𝑣𝑒𝑟𝑠𝑎𝑙 ∶ 𝑡𝑎𝑘𝑒 𝑚𝑖𝑟𝑟𝑜𝑟 𝑖𝑚𝑎𝑔𝑒 𝑎𝑏𝑜𝑢𝑡 𝑌 𝑎𝑥𝑖𝑠
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Find x(-t) for each of the example:
Example 2
Example 3
Value on amplitude remains same.
Fold the signal along ySNSaxis
Example 4
Time Scaling of Signals
•
•
•
•
If x( t ) is a original signal.
Then y(t) = x(αt) is time scaled version of the signal x(t), there α is always positive.
If | α | > 1 ---- leads to compression of signal on time axis (speeds up the signal)
If | α | < 1 ----- leads to expansion of signal on time axis (slows down the signal)
𝑁𝑜𝑡𝑒: 𝑇ℎ𝑒 𝑎𝑚𝑝𝑙𝑖𝑡𝑢𝑑𝑒 𝑜𝑓SNS𝑡ℎ𝑒 𝑠𝑖𝑔𝑛𝑎𝑙 𝑟𝑒𝑚𝑎𝑖𝑛𝑠 𝑠𝑎𝑚𝑒
Meaning of time scaling
Consider that signal x(t) represents a tape recording
X(2t) is same as record played at twice the speed (therefore will run faster in time
-hence compressed)
X(t/2) is same recording played at half the speed (therefore will run slower in time
-hence expanded)
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Example 2 (Time Scaling of Signals)
Find x(2t)
Find x(t/2)
Expanded
Compressed
Note : that time is scaled amplitude remains same
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Example 3
Find x(2t)
Find x(t/2)
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Exercise: Time scaling application
• Go to any you tube video
• Go to settings
• Go to playback speed
• Adjust to 0.75 and check what happens
• Adjust to 1.25 and check what happens
α = 0.75, 𝑝𝑙𝑎𝑦𝑒𝑑 𝑠𝑙𝑜𝑤 … 𝑡ℎ𝑎𝑡 𝑖𝑠 𝑠𝑖𝑔𝑛𝑎𝑙 𝑖𝑠 𝑒𝑥𝑝𝑎𝑛𝑑𝑒𝑑 𝑥 α𝑡 𝑤ℎ𝑒𝑟𝑒 α < 1
α = 1.25, 𝑝𝑙𝑎𝑦𝑒𝑑 𝑓𝑎𝑠𝑡 … 𝑡ℎ𝑎𝑡 𝑖𝑠 𝑠𝑖𝑔𝑛𝑎𝑙 𝑖𝑠 𝑐𝑜𝑚𝑝𝑟𝑒𝑠𝑠𝑒𝑑 𝑥 α𝑡 𝑤ℎ𝑒𝑟𝑒 α > 1
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Time shifting of signals
• Time shifting is mathematically expressed as,
Where, X(t) is the original signal, and t0 represents the shift in time.
Note: If shift t0 > 0 (+ve)…………the signal is said to be right shifted (delayed) → x(t - t0)
If t0 < 0 (-ve) ………….implies the signal is left shifted (advanced) → x(t + t0)
Its generally used to fast-forward (jump in time)
or delay a signal (go back in time)
as is necessary in most practical circumstances.
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Example
Note: If shift t0 > 0 (+ve) x(t- t0 ) → The signal is shifted to right by t0 (delayed)
→ move the origin of x(t) to to
Meaning → the signal is the signal x(t) that was t0 seconds ago
If t0 < 0 (-ve) ………….implies the signal is left shifted by t0 (advanced).
Find x(t+4)
Find x(t-3)
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Why x(t - to) signal shift to right and
x(t + to) signal shift to left
Say a signal was launched at x(t)
x(t - to)
x ( t – (- to))
x ( t + to))
Signal is delayed by to
Signal is advanced
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Basic Signal Operations
I.
The following operation can be performed with amplitude:
Amplitude Scaling: a.x(t)
Amplitude Reversal : - x(t)
Addition of signals: x1(t) + x2(t)
Subtraction of signals : x1(t) - x2(t)
Multiplication of signals : x1(t) . x2(t)
II. The following operations can be performed with time:
Convolution of signals : x1(t) * x2(t)
Time Shifting:
x(t – to)
Time Scaling:
x(at)
Time Reversal : x ( - t )
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Examples :
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Construction of different signals
• Using elementary signals we can construct different kinds of signal
• Elementary signals
(i)
Unit step signal, u(t)
(ii)
Unit impulse signal, δ(t)
(iii) Ramp signal, r(t)
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y(t) = u (t) - u (t-2)
Sketch the signal y(t) = u (t) - u (t-2)
y(t) = u (t) - u (t-2)
y(t) = u (t) - u (t-2)
3. Plot -u (t-2)
1. Plot u (t)
y(t) = u (t) - u (t-2)
2. Plot u (t-2)
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Sketch the signal y(t) = u (t) . u (t-2)
Multiply the two signals
1. Plot u (t)
2. Plot u (t-2)
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Let is now see the summary of all operations
1. Plot x(t) = u (t) - u (t-2)
2. Plot y(t) = x (2t)
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Sketch x(t) = u (t) - u (t-2) and plot y(t)=x(t/2)
1. Plot x(t) = u (t) - u (t-2)
2. Plot x(t/2)
i.e expand the time scale by 2
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Plot y(t) = u (2t) - u (2t-2)
a. Do shifting first (not mandatory but easier step)
1. Plot u (2t))
b. Then do scaling
2. - u (2t-2)
c. Amplitude reversal
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Sketch u (t/2) - u (t/2 - 2)
1. Plot u (t)
3. Plot - u (t/2 - 2)
a. Plot u (t )
c. Plot u (t/2 - 2 )
b. Plot u (t - 2 )
d. Plot - u (t/2 - 2 )
2. Plot u (t/2)
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Plot the following for the given signal
Find x(t+2)
Find x(t-2)
Shift right by 2 units
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Example
𝐶𝑜𝑚𝑝𝑢𝑡𝑒: 𝑥(2𝑡 + 3)
𝑆𝑡𝑒𝑝 2: 𝐷𝑜 𝑡𝑖𝑚𝑒 𝑠𝑐𝑎𝑙𝑖𝑛𝑔
𝑥(2𝑡 + 3)
𝐶𝑜𝑚𝑝𝑟𝑒𝑠𝑠 𝑡ℎ𝑒 𝑠𝑖𝑔𝑛𝑎𝑙 𝑥 𝑡 + 3 𝑏𝑦 2 𝑢𝑛𝑖𝑡𝑠
𝑆𝑡𝑒𝑝 1 ∶ 𝐷𝑜 𝑡𝑖𝑚𝑒𝑠ℎ𝑖𝑓𝑖𝑡𝑖𝑛𝑔 ∶ 𝑥(𝑡 + 3)
𝑆ℎ𝑖𝑓𝑡 𝑠𝑖𝑔𝑛𝑎𝑙 𝑡𝑜 𝑙𝑒𝑓𝑡 𝑏𝑦 3 𝑢𝑛𝑖𝑡𝑠
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Example
2. Compress by factor 2
𝐶𝑜𝑚𝑝𝑢𝑡𝑒: 𝑥(−2𝑡 + 3)
𝑥(−2𝑡 + 3)
3. Time reversal
𝑆𝑡𝑒𝑝 1 ∶ 𝐷𝑜 𝑡𝑖𝑚𝑒𝑠ℎ𝑖𝑓𝑖𝑡𝑖𝑛𝑔 ∶ 𝑥(𝑡 + 3)
𝑆ℎ𝑖𝑓𝑡 𝑠𝑖𝑔𝑛𝑎𝑙 𝑡𝑜 𝑙𝑒𝑓𝑡 𝑏𝑦 3 𝑢𝑛𝑖𝑡𝑠
Order of compression/ expansion/Time reversal does not matter
Time shifting always has to be done first
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Why do we do shift operation first?
u (2t-2)
1. If compression x(2t) or expansion x(t/2) is carries prior to the shift
operation, then shift also needs to be compressed / expanded by same
factor
Therefore, to avoid confusion while solving problems, students are advised to
carry the shift operation (if it exist) of x(t) first and then proceed with the
reversal/compression/expansion etc.
This practice will enable to solve questions easily
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