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Notes based on the Signals and Systems course from the MIT Open Courseware (OCW) site.

Jerusalem College of Technology

Signals and SystemsLecture #2-3

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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

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• 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

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• 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

Representation of DT Signals Using Unit Samples

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That is ...

Coefficients Basic Signals

The Sifting Property of the Unit Sample

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• 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:

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• 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:

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Convolution Sum Representation of

Response of LTI Systems

Interpretation

n n

n n

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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

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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

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0 × 1 + 1 ×2 = 2

(-1) × 1 + 0 × 2 + 1 × (-1) = -2

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• 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

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• 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)

• Approximate any input x(t) as a sum of shifted, scaled

pulses

Representation of CT Signals

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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)

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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

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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

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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()

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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-)

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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-)

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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-)

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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()

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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

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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)

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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]

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DISTRIBUTIVITY (CT and DT) LTI

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ASSOCIATIVITY (CT and DT) LTI

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LTI Causality and Stability

k

kh ][Stability: DT LTI system is stable ↔

Causality: DT LTI system is causal↔ h[n] = 0, n<0

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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.

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The Unit Doublet — Differentiator

Impulse response = unit doublet

The operational definition of the unit doublet:

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Triplets and beyond!

n is number of

differentiations

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Integrators

―-1 derivatives" = integral I.R. = unit step

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Integrators (continued)

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Notation

Define

Then

E.g.

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Sometimes Useful Tricks

Differentiate first, then convolve, then integrate

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Example

1 2-1

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Example (continued)

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