Jerusalem College of Technology Signals and...
Transcript of Jerusalem College of Technology Signals and...
![Page 1: Jerusalem College of Technology Signals and Systemshomedir.jct.ac.il/~echalom/courses/130601-2011b/lecture_02-03.pdf · Notes based on the Signals and Systems course from the MIT](https://reader034.fdocuments.in/reader034/viewer/2022042119/5e98af4cb9cecb6b2371118b/html5/thumbnails/1.jpg)
1
Notes based on the Signals and Systems course from the MIT Open Courseware (OCW) site.
Jerusalem College of Technology
Signals and SystemsLecture #2-3
![Page 2: Jerusalem College of Technology Signals and Systemshomedir.jct.ac.il/~echalom/courses/130601-2011b/lecture_02-03.pdf · Notes based on the Signals and Systems course from the MIT](https://reader034.fdocuments.in/reader034/viewer/2022042119/5e98af4cb9cecb6b2371118b/html5/thumbnails/2.jpg)
2
1. Discrete Time (DT) Signals and Systems
a. DT Unit Sample / Impulse and Unit Step Functions
b. DT signal represented as shifted, weighted samples
c. DT Convolution sum for LTI system
d. DT Impulse Response
2. Continuous Time (CT) Signals and Systems
a. CT Unit Step Functions and Unit Impulse Function
b. CT signal represented as shifted, weighted impulses
c. CT Convolution Integral for LTI system
d. CT Impulse Response
3. Convolution Properties: Commutative, Distributive,
Associative, Causal, Stable, Memoryless, etc.
OVERVIEW
![Page 3: Jerusalem College of Technology Signals and Systemshomedir.jct.ac.il/~echalom/courses/130601-2011b/lecture_02-03.pdf · Notes based on the Signals and Systems course from the MIT](https://reader034.fdocuments.in/reader034/viewer/2022042119/5e98af4cb9cecb6b2371118b/html5/thumbnails/3.jpg)
3
• We define the DT unit sample function, δ[n], as:
• Note that
UNIT SAMPLE FUNCTION
0,0
0,1][
n
nn
-5 0 50
0.2
0.4
0.6
0.8
1
n
[n
]
DT Unit Sample Function
… …
1][
n
n
![Page 4: Jerusalem College of Technology Signals and Systemshomedir.jct.ac.il/~echalom/courses/130601-2011b/lecture_02-03.pdf · Notes based on the Signals and Systems course from the MIT](https://reader034.fdocuments.in/reader034/viewer/2022042119/5e98af4cb9cecb6b2371118b/html5/thumbnails/4.jpg)
4
• We define the DT unit step function, u[n], as:
• Note: δ[n] = u[n] – u[n-1]
and
UNIT STEP FUNCTION
0,0
0,1][
n
nnu
-5 0 50
0.2
0.4
0.6
0.8
1
n
u[n
]
DT Unit Step Function
…
…
0
][][][k
n
k
knknu
![Page 5: Jerusalem College of Technology Signals and Systemshomedir.jct.ac.il/~echalom/courses/130601-2011b/lecture_02-03.pdf · Notes based on the Signals and Systems course from the MIT](https://reader034.fdocuments.in/reader034/viewer/2022042119/5e98af4cb9cecb6b2371118b/html5/thumbnails/5.jpg)
Representation of DT Signals Using Unit Samples
5
![Page 6: Jerusalem College of Technology Signals and Systemshomedir.jct.ac.il/~echalom/courses/130601-2011b/lecture_02-03.pdf · Notes based on the Signals and Systems course from the MIT](https://reader034.fdocuments.in/reader034/viewer/2022042119/5e98af4cb9cecb6b2371118b/html5/thumbnails/6.jpg)
That is ...
Coefficients Basic Signals
The Sifting Property of the Unit Sample
6
![Page 7: Jerusalem College of Technology Signals and Systemshomedir.jct.ac.il/~echalom/courses/130601-2011b/lecture_02-03.pdf · Notes based on the Signals and Systems course from the MIT](https://reader034.fdocuments.in/reader034/viewer/2022042119/5e98af4cb9cecb6b2371118b/html5/thumbnails/7.jpg)
• Recall that if a system is linear it obeys superposition:
If x1[n] y1[n] and x2[n] y2[n]
then a1x1[n] + a2x2[n] a1y1[n] + a2y2[n]
• Now suppose the system above is linear, and we define
hk[n] as the response (output) to [n - k]:
• Meaning:
• Then From superposition:
7
![Page 8: Jerusalem College of Technology Signals and Systemshomedir.jct.ac.il/~echalom/courses/130601-2011b/lecture_02-03.pdf · Notes based on the Signals and Systems course from the MIT](https://reader034.fdocuments.in/reader034/viewer/2022042119/5e98af4cb9cecb6b2371118b/html5/thumbnails/8.jpg)
• Now suppose the system is LTI, and we define the unit
sample response h[n] as the output of a unit sample
input:
From LTI:
From TI:
8
![Page 9: Jerusalem College of Technology Signals and Systemshomedir.jct.ac.il/~echalom/courses/130601-2011b/lecture_02-03.pdf · Notes based on the Signals and Systems course from the MIT](https://reader034.fdocuments.in/reader034/viewer/2022042119/5e98af4cb9cecb6b2371118b/html5/thumbnails/9.jpg)
Convolution Sum Representation of
Response of LTI Systems
Interpretation
n n
n n
9
![Page 10: Jerusalem College of Technology Signals and Systemshomedir.jct.ac.il/~echalom/courses/130601-2011b/lecture_02-03.pdf · Notes based on the Signals and Systems course from the MIT](https://reader034.fdocuments.in/reader034/viewer/2022042119/5e98af4cb9cecb6b2371118b/html5/thumbnails/10.jpg)
Visualizing the calculation of
y[0] = prod of
overlap for
n = 0
y[1] = prod of
overlap for
n = 1
Choose value of n and consider it fixed
View as functions of k with n fixed
10
![Page 11: Jerusalem College of Technology Signals and Systemshomedir.jct.ac.il/~echalom/courses/130601-2011b/lecture_02-03.pdf · Notes based on the Signals and Systems course from the MIT](https://reader034.fdocuments.in/reader034/viewer/2022042119/5e98af4cb9cecb6b2371118b/html5/thumbnails/11.jpg)
Calculating Successive Values: Shift, Multiply, Sum
-1
1 × 1 = 1
(-1) × 2 + 0 × (-1) + 1 × (-1) = -3
(-1) × (-1) + 0 × (-1) = 1
(-1) × (-1) = 1
4
0 × 1 + 1 ×2 = 2
(-1) × 1 + 0 × 2 + 1 × (-1) = -2
11
![Page 12: Jerusalem College of Technology Signals and Systemshomedir.jct.ac.il/~echalom/courses/130601-2011b/lecture_02-03.pdf · Notes based on the Signals and Systems course from the MIT](https://reader034.fdocuments.in/reader034/viewer/2022042119/5e98af4cb9cecb6b2371118b/html5/thumbnails/12.jpg)
12
• We define the CT unit step function, u(t), as:
• Note that u(t) is discontinuous at t = 0.
CONTINUOUS TIME UNIT STEP FUNCTION
0,0
0,1)(
t
ttu
…
…
-3 -2 -1 0 1 2 30
0.2
0.4
0.6
0.8
1
t
u(t
)
CT Unit Step Function
![Page 13: Jerusalem College of Technology Signals and Systemshomedir.jct.ac.il/~echalom/courses/130601-2011b/lecture_02-03.pdf · Notes based on the Signals and Systems course from the MIT](https://reader034.fdocuments.in/reader034/viewer/2022042119/5e98af4cb9cecb6b2371118b/html5/thumbnails/13.jpg)
13
• The unit step function, u(t), is the running integral of
the unit impulse function, δ(t).
• Therefore the unit impulse function is the derivative of
the unit step function.
CONTINUOUS TIME UNIT IMPULSE FUNCTION
dt
tdut
dttu
dtu
tdt
t
t
)()(
)()(
)()(
1)(
0
……
Note that Since u(t) is discontinuous at t = 0,
it is not formally differentiable
-3 -2 -1 0 1 2 30
0.2
0.4
0.6
0.8
1
t
(t
)
CT Unit Impulse Function
(1)
![Page 14: Jerusalem College of Technology Signals and Systemshomedir.jct.ac.il/~echalom/courses/130601-2011b/lecture_02-03.pdf · Notes based on the Signals and Systems course from the MIT](https://reader034.fdocuments.in/reader034/viewer/2022042119/5e98af4cb9cecb6b2371118b/html5/thumbnails/14.jpg)
• Approximate any input x(t) as a sum of shifted, scaled
pulses
Representation of CT Signals
14
![Page 15: Jerusalem College of Technology Signals and Systemshomedir.jct.ac.il/~echalom/courses/130601-2011b/lecture_02-03.pdf · Notes based on the Signals and Systems course from the MIT](https://reader034.fdocuments.in/reader034/viewer/2022042119/5e98af4cb9cecb6b2371118b/html5/thumbnails/15.jpg)
otherwise
tt
,0
0,1
)(
δΔ(t) has area = 1
δΔ(t-k Δ)Δ has
amplitude = 1
The Sifting Property of the Unit Impulse
Approximation to CT Impulse Function
Note that in the limit as Δ→0, δΔ(t) →δ(t)
15
![Page 16: Jerusalem College of Technology Signals and Systemshomedir.jct.ac.il/~echalom/courses/130601-2011b/lecture_02-03.pdf · Notes based on the Signals and Systems course from the MIT](https://reader034.fdocuments.in/reader034/viewer/2022042119/5e98af4cb9cecb6b2371118b/html5/thumbnails/16.jpg)
Response of a CT LTI System
• Suppose the input x(t) = δΔ(t) results in an output signal
hΔ(t) then if we define a new input signal,
• In the limit that Δ→0, the summation becomes an integral
• h(t) is the impulse response, meaning that if the input is an
impulse, δ(t), then the output is h(t).
)()()(ˆ)()()(ˆ kthkxtyktkxtxkk
dthxtydtxtx )()()()()()(
Convolution Integral
16
![Page 17: Jerusalem College of Technology Signals and Systemshomedir.jct.ac.il/~echalom/courses/130601-2011b/lecture_02-03.pdf · Notes based on the Signals and Systems course from the MIT](https://reader034.fdocuments.in/reader034/viewer/2022042119/5e98af4cb9cecb6b2371118b/html5/thumbnails/17.jpg)
Example: CT convolution
Operation of CT Convolution
h(t) = t+2, for -2≤t≤-1
= 0, elsewhere
h(t-τ) = t-τ+2, for -2≤t-τ≤-1
i.e. h(t-τ) = - τ+t+2, for t+1≤τ≤t+2
x(t) = 1, for 1≤t≤3
= 0, elsewhere
x(τ) = 1, for 1≤ τ ≤3
= 0, elsewhere
17
![Page 18: Jerusalem College of Technology Signals and Systemshomedir.jct.ac.il/~echalom/courses/130601-2011b/lecture_02-03.pdf · Notes based on the Signals and Systems course from the MIT](https://reader034.fdocuments.in/reader034/viewer/2022042119/5e98af4cb9cecb6b2371118b/html5/thumbnails/18.jpg)
Convolution Example: t ≤ -1
Time Interval x(τ)·h(t-τ) Overlap Interval y(t)
t≤-1 0 None 0
-5 -4 -3 t=-2 t+1 t+2 1 2 3 4 50
0.2
0.4
0.6
0.8
1
Convolution Example, t-1
h(t-)
x()
18
![Page 19: Jerusalem College of Technology Signals and Systemshomedir.jct.ac.il/~echalom/courses/130601-2011b/lecture_02-03.pdf · Notes based on the Signals and Systems course from the MIT](https://reader034.fdocuments.in/reader034/viewer/2022042119/5e98af4cb9cecb6b2371118b/html5/thumbnails/19.jpg)
Convolution Example: -1 ≤ t ≤ 0
Time Interval x(τ)·h(t-τ) Overlap Interval y(t)
-1≤t≤0 -τ+t+2 1≤τ≤t+2
2
)1(
)2(
2
2
1
t
dt
t
-5 -4 -3 -2 -1t=-0.3 t+1 t+2 3 4 50
0.5
1
Convolution Example, -1t0
h(t-)
x()
-5 -4 -3 -2 -1 0 1 t+2 3 4 50
0.5
t+1
1
x()h(t-)
19
![Page 20: Jerusalem College of Technology Signals and Systemshomedir.jct.ac.il/~echalom/courses/130601-2011b/lecture_02-03.pdf · Notes based on the Signals and Systems course from the MIT](https://reader034.fdocuments.in/reader034/viewer/2022042119/5e98af4cb9cecb6b2371118b/html5/thumbnails/20.jpg)
Convolution Example: 0 ≤ t ≤ 1
Time Interval x(τ)·h(t-τ) Overlap Interval y(t)
0≤t≤1 -τ+t+2 t+1≤τ≤t+2
2
1
)2(
2
1
dt
t
t
-5 -4 -3 -2 -1 t=0.3 t+1 t+2 3 4 50
0.5
1
Convolution Example, 0t1
h(t-)
x()
-5 -4 -3 -2 -1 0 t+1 t+2 3 4 50
0.5
1
x()h(t-)
20
![Page 21: Jerusalem College of Technology Signals and Systemshomedir.jct.ac.il/~echalom/courses/130601-2011b/lecture_02-03.pdf · Notes based on the Signals and Systems course from the MIT](https://reader034.fdocuments.in/reader034/viewer/2022042119/5e98af4cb9cecb6b2371118b/html5/thumbnails/21.jpg)
Convolution Example: 1 ≤ t ≤ 2
Time Interval x(τ)·h(t-τ) Overlap Interval y(t)
1≤t≤2 -τ+t+2 t+1≤τ≤3
2
)1(
2
1
)2(
2
3
1
t
dtt
-5 -4 -3 -2 -1 0 t=1.3 t+1 t+2 4 50
0.5
1
Convolution Example, 1t2
h(t-)
x()
-5 -4 -3 -2 -1 0 1 t+1 3 4 50
t-1
0.5
1
x()h(t-)
21
![Page 22: Jerusalem College of Technology Signals and Systemshomedir.jct.ac.il/~echalom/courses/130601-2011b/lecture_02-03.pdf · Notes based on the Signals and Systems course from the MIT](https://reader034.fdocuments.in/reader034/viewer/2022042119/5e98af4cb9cecb6b2371118b/html5/thumbnails/22.jpg)
Convolution Example: t ≥ 2
Time Interval x(τ)·h(t-τ) Overlap Interval y(t)
t≥2 0 None 0
-5 -4 -3 -2 -1 0 1 2 t=3 t+1 t+20
0.2
0.4
0.6
0.8
1
Convolution Example, t2
h(t-)
x()
22
![Page 23: Jerusalem College of Technology Signals and Systemshomedir.jct.ac.il/~echalom/courses/130601-2011b/lecture_02-03.pdf · Notes based on the Signals and Systems course from the MIT](https://reader034.fdocuments.in/reader034/viewer/2022042119/5e98af4cb9cecb6b2371118b/html5/thumbnails/23.jpg)
Convolution Example:
-5 -4 -3 -2 -1 0 1 2 3 4 50
0.1
0.2
0.3
0.4
0.5
t
y(t
)Convolution Output
23
![Page 24: Jerusalem College of Technology Signals and Systemshomedir.jct.ac.il/~echalom/courses/130601-2011b/lecture_02-03.pdf · Notes based on the Signals and Systems course from the MIT](https://reader034.fdocuments.in/reader034/viewer/2022042119/5e98af4cb9cecb6b2371118b/html5/thumbnails/24.jpg)
CT LTI PROPERTIES AND EXAMPLES
1) Commutativity:
2)
3) An integrator:
impulse response of
system is h(t) = u(t)
4) Step response: input x(t) = u(t)
24
![Page 25: Jerusalem College of Technology Signals and Systemshomedir.jct.ac.il/~echalom/courses/130601-2011b/lecture_02-03.pdf · Notes based on the Signals and Systems course from the MIT](https://reader034.fdocuments.in/reader034/viewer/2022042119/5e98af4cb9cecb6b2371118b/html5/thumbnails/25.jpg)
DT LTI PROPERTIES AND EXAMPLES
1) Commutativity: x[n]*h[n] = h[n]*x[n]
2) x[n]*δ[n-N]=x[n-N] Sifting property: x[n]*δ[n] = x[n]
3) An accumulator:
That is:
4) Step response: input x[n]=u[n]
][][][
][][
nuknh
kxny
n
k
n
k
So if input x[n] = δ[n]
output y[n] = h[n]
n
k
kxnunxnhnxny ][][*][][*][][
n
k
khnunhnhnuns ][][*][][*][][
impulse response of
system is h[n] = u[n]
25
![Page 26: Jerusalem College of Technology Signals and Systemshomedir.jct.ac.il/~echalom/courses/130601-2011b/lecture_02-03.pdf · Notes based on the Signals and Systems course from the MIT](https://reader034.fdocuments.in/reader034/viewer/2022042119/5e98af4cb9cecb6b2371118b/html5/thumbnails/26.jpg)
DISTRIBUTIVITY (CT and DT) LTI
26
![Page 27: Jerusalem College of Technology Signals and Systemshomedir.jct.ac.il/~echalom/courses/130601-2011b/lecture_02-03.pdf · Notes based on the Signals and Systems course from the MIT](https://reader034.fdocuments.in/reader034/viewer/2022042119/5e98af4cb9cecb6b2371118b/html5/thumbnails/27.jpg)
ASSOCIATIVITY (CT and DT) LTI
27
![Page 28: Jerusalem College of Technology Signals and Systemshomedir.jct.ac.il/~echalom/courses/130601-2011b/lecture_02-03.pdf · Notes based on the Signals and Systems course from the MIT](https://reader034.fdocuments.in/reader034/viewer/2022042119/5e98af4cb9cecb6b2371118b/html5/thumbnails/28.jpg)
LTI Causality and Stability
k
kh ][Stability: DT LTI system is stable ↔
Causality: DT LTI system is causal↔ h[n] = 0, n<0
28
![Page 29: Jerusalem College of Technology Signals and Systemshomedir.jct.ac.il/~echalom/courses/130601-2011b/lecture_02-03.pdf · Notes based on the Signals and Systems course from the MIT](https://reader034.fdocuments.in/reader034/viewer/2022042119/5e98af4cb9cecb6b2371118b/html5/thumbnails/29.jpg)
The Operational Definition of the Unit Impulse (t)
δ(t) — idealization of a unit-area pulse that is so short that, for
any physical systems of interest to us, the system responds
only to the area of the pulse and is insensitive to its duration
Operationally: The unit impulse is the signal which when
applied to any LTI system results in an output equal to the
impulse response of the system. That is,
— δ(t) is defined by what it does under convolution.
29
![Page 30: Jerusalem College of Technology Signals and Systemshomedir.jct.ac.il/~echalom/courses/130601-2011b/lecture_02-03.pdf · Notes based on the Signals and Systems course from the MIT](https://reader034.fdocuments.in/reader034/viewer/2022042119/5e98af4cb9cecb6b2371118b/html5/thumbnails/30.jpg)
The Unit Doublet — Differentiator
Impulse response = unit doublet
The operational definition of the unit doublet:
30
![Page 31: Jerusalem College of Technology Signals and Systemshomedir.jct.ac.il/~echalom/courses/130601-2011b/lecture_02-03.pdf · Notes based on the Signals and Systems course from the MIT](https://reader034.fdocuments.in/reader034/viewer/2022042119/5e98af4cb9cecb6b2371118b/html5/thumbnails/31.jpg)
Triplets and beyond!
n is number of
differentiations
31
![Page 32: Jerusalem College of Technology Signals and Systemshomedir.jct.ac.il/~echalom/courses/130601-2011b/lecture_02-03.pdf · Notes based on the Signals and Systems course from the MIT](https://reader034.fdocuments.in/reader034/viewer/2022042119/5e98af4cb9cecb6b2371118b/html5/thumbnails/32.jpg)
Integrators
―-1 derivatives" = integral I.R. = unit step
32
![Page 33: Jerusalem College of Technology Signals and Systemshomedir.jct.ac.il/~echalom/courses/130601-2011b/lecture_02-03.pdf · Notes based on the Signals and Systems course from the MIT](https://reader034.fdocuments.in/reader034/viewer/2022042119/5e98af4cb9cecb6b2371118b/html5/thumbnails/33.jpg)
Integrators (continued)
33
![Page 34: Jerusalem College of Technology Signals and Systemshomedir.jct.ac.il/~echalom/courses/130601-2011b/lecture_02-03.pdf · Notes based on the Signals and Systems course from the MIT](https://reader034.fdocuments.in/reader034/viewer/2022042119/5e98af4cb9cecb6b2371118b/html5/thumbnails/34.jpg)
Notation
Define
Then
E.g.
34
![Page 35: Jerusalem College of Technology Signals and Systemshomedir.jct.ac.il/~echalom/courses/130601-2011b/lecture_02-03.pdf · Notes based on the Signals and Systems course from the MIT](https://reader034.fdocuments.in/reader034/viewer/2022042119/5e98af4cb9cecb6b2371118b/html5/thumbnails/35.jpg)
Sometimes Useful Tricks
Differentiate first, then convolve, then integrate
35
![Page 36: Jerusalem College of Technology Signals and Systemshomedir.jct.ac.il/~echalom/courses/130601-2011b/lecture_02-03.pdf · Notes based on the Signals and Systems course from the MIT](https://reader034.fdocuments.in/reader034/viewer/2022042119/5e98af4cb9cecb6b2371118b/html5/thumbnails/36.jpg)
Example
1 2-1
36
![Page 37: Jerusalem College of Technology Signals and Systemshomedir.jct.ac.il/~echalom/courses/130601-2011b/lecture_02-03.pdf · Notes based on the Signals and Systems course from the MIT](https://reader034.fdocuments.in/reader034/viewer/2022042119/5e98af4cb9cecb6b2371118b/html5/thumbnails/37.jpg)
Example (continued)
37