Superposition and Convolution
Transcript of Superposition and Convolution
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6.003: Signal Processing
Superposition and Convolution
y[n] = (h ∗ x)[n] =�
m
h[m]x[n − m]
6 October 2020
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Systems
So far, we have focused our attention on signals and their math-
ematical representations. However, we are often also interested in
manipulating signals. To this end, we introduce the notion of a
system (or filter).
systemsignal
in
signal
out
Examples:
• audio enhancement: equalization, noise reduction, reverberation,
echo cancellation, pitch shift (auto-tune)
• image enhancement: smoothing, edge enhancement, unsharp
masking, feature detection
• video enhancement: image stabilization, motion magnification
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Representations of Signals
Characterize systems by their input/output relationships.
systemsignal
in
signal
out
Difference Equation: represent system by algebraic constraints on
samples
Convolution: represent system by its unit sample response
Filter: represent system as amplfication/attentuation of frequency
components
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Linearity
A system is linear if its response to a weighted sum of inputs is
equal to the weighted sum of its responses to each of the inputs.
Given
systemx1[n] y1[n]
and
systemx2[n] y2[n]
the system is linear if
systemαx1[n] + βx2[n] αy1[n] + βy2[n]
is true for all α and β and all possible inputs.
A system is linear if it is both additive and homogeneous.
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Time-Invariance
A system is time-invariant if delaying the input to the system simply
delays the output by the same amount of time.
Given
systemx[n] y[n]
the system is time invariant if
systemx[n−n0] y[n−n0]
is true for all n0 and for all possible inputs.
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Unit Sample Response
If a system is linear and time-invariant, its input-output relation is
completely specified by the system’s unit sample response h[n].
systemx[n] y[n]
The unit sample response h[n] is the output of the system when the
input is the unit sample signal δ[n].
systemδ[n] h[n]
The output for more complicated inputs can be computed by sum-
ming scaled and shifted versions of the unit sample response.
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Superposition
Consider the following signal:
x[n] =
1 if n = 0−1 if n = 3−2 if n = 40 otherwise
This signal can be represented as:
x[n] = δ[n] − δ[n − 3] − 2δ[n − 4]
In general, we can represent a signal as a sum of scaled, shifted
deltas:
x[n] =∞�
m=−∞x[m]δ[n − m]
= . . . + x[−1]δ[n + 1] + x[0]δ[n] + x[1]δ[n − 1] + x[2]δ[n − 2] + . . .
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Superposition
In general, we can represent a signal as a sum of scaled, shifted
deltas:
x[n] =∞�
m=−∞x[m]δ[n − m]
= . . . + x[−1]δ[n + 1] + x[0]δ[n] + x[1]δ[n − 1] + x[2]δ[n − 2] + . . .
If h[·] is the unit sample response of an LTI system, then the output
of that system in response to this arbitrary input x[·] can be viewed
as a sum of scaled, shifted unit sample responses:
y[n] =∞�
m=−∞x[m]h[n − m]
= . . . + x[−1]h[n + 1] + x[0]h[n] + x[1]h[n − 1] + x[2]h[n − 2] + . . .
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Structure of Superposition
If a system is linear and time-invariant (LTI) then its output is the
sum of weighted and shifted unit sample responses.
systemδ[n] h[n]
systemδ[n − m] h[n − m]
systemx[m]δ[n − m] x[m]h[n − m]
systemx[n] =∞�
m=−∞x[m]δ[n − m] y[n] =
∞�
m=−∞x[m]h[n − m]
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Superposition by Example
Consider an LTI system described by y[n] = 3x[n] − x[n − 1]. Compute
the response of this system to an input x[·] given by:
x[n] = 2δ[n − 1] + 5δ[n − 2] + 3δ[n − 4]
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Convolution
Response of an LTI system to an arbitrary input.
LTIx[n] y[n]
y[n] =∞�
m=−∞x[m]h[n − m] ≡ (x ∗ h)[n]
This operation is called convolution (verb form: convolve).
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Convolution
Convolution is represented with an asterisk.∞�
m=−∞x[m]h[n − m] ≡ (x ∗ h)[n]
Convolution operates on signals, not samples.
The symbols x and h represent DT signals.
Convolving x with h generates a new DT signal x ∗ h.
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DT Convolution: Summary
Unit sample response h[n] is a complete description of an LTI system.
h[n]x[n] y[n]
Given h[·], we can compute the response y[·] to any input x[·] by
convolving x[·] and h[·]:
y[n] = (x ∗ h)[n] ≡∞�
m=−∞x[m]h[n − m]
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Discrete-Time Example: Solera Process
Aging and blending wines from different crops.
Start with 3 barrels of wine: newest at left, oldest at right.
−1 −2 −3
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Discrete-Time Example: Solera Process
Aging and blending wines from different crops.
Sell half of the oldest stock.
−1 −2 −3
−3sell
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Discrete-Time Example: Solera Process
Aging and blending wines from different crops.
Refill oldest barrel from next-to-oldest barrel.
−1 −2−2−3
−3sell
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Discrete-Time Example: Solera Process
Aging and blending wines from different crops.
Refill next-to-oldest barrel from youngest barrel.
−1−1−2
−2−3
−3sell
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Discrete-Time Example: Solera Process
Aging and blending wines from different crops.
Refill youngest barrel with this year’s harvest.
0 0−1
−1−2
−2−3
−3sell
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Discrete-Time Example: Solera Process
Aging and blending wines from different crops.
Old and new contents mix; ready for next year.
0−1
−1−2
−2−3
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Discrete-Time Example: Solera Process
Aging and blending wines from different crops.
Old and new contents mix; ready for next year.
0−1
−1−2
−2−3
Properties of solera process:
• Mixing produces a more uniform product.
• Mitigates worst-case results of one bad year.
• Blends wines from MANY previous years.
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Solera Analysis
We can analyze these effects with a tracer experiment.
Add 1 unit of tracer to new crop; track tracer through the system.
1 0 0year 0
12
12 0year 1
14
12
14year 2
How much tracer will be in each barrel at the end of year 3?
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Solera Analysis
Add 1 unit of tracer to new crop; track tracer through the system.
Year Tracer in Barrel Barrel Barrel Tracer outn x[n] #1 #2 #3 y[n]0 1 1 0 0 01 0 1/2 1/2 0 02 0 1/4 2/4 1/4 03 0 1/8 3/8 3/8 1/84 0 1/16 4/16 6/16 3/165 0 1/32 5/32 10/32 6/326 0 1/64 6/64 15/64 10/64
1
n
x[n]3/16
n
y[n]
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Solera Analysis
How would results change if tracer were added in year 1 (not 0)?
Original response:
1
n
x[n]3/16
n
y[n]
Delayed input → delayed output:
1
n
x[n − 1]3/16
n
y[n − 1]
Delaying the input by a year simply delays the outputs by one year.
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Time-Invariance
A system is time-invariant if delaying the input to the system simply
delays the output by the same amount of time.
Given
systemx[n] y[n]
the system is time invariant if
systemx[n − n0] y[n − n0]
is true for all n0.
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Solera Analysis
Scaling the input amplitudes:
1
n
0.5x[n]3/16
n
0.5y[n]
Adding two inputs:
1
n
x[n] + x[n − 6]3/16
n
y[n] + y[n − 6]
Linearly combining two inputs:
1
n
x[n] + 0.5x[n − 6]3/16
n
y[n] + 0.5y[n − 6]
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Linearity
A system is linear if its response to a weighted sum of inputs is equal
to the weighted sum of its responses to each of the inputs.
Given
systemx1[n] y1[n]
and
systemx2[n] y2[n]
the system is linear if
systemαx1[n] + βx2[n] αy1[n] + βy2[n]
is true for all α and β.
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Convolution
If a system is linear and time invariant, the response to an arbitrary
input is the convolution of that input with the unit-sample response.
systemδ[n] h[n]
systemδ[n − m] h[n − m]
systemx[k]δ[n − m] x[k]h[n − m]
systemx[n] =∞�
m=−∞x[k]δ[n − m] y[n] =
∞�
m=−∞x[k]h[n − m]
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Check Yourself
For solera process ...
Year Tracer in Barrel Barrel Barrel Tracer outn x[n] #1 #2 #3 y[n]0 1 1 0 0 01 0 1/2 1/2 0 02 0 1/4 2/4 1/4 03 0 1/8 3/8 3/8 1/84 0 1/16 4/16 6/16 3/165 0 1/32 5/32 10/32 6/326 0 1/64 6/64 15/64 10/64
How much tracer would be found in output from year 5,
if one unit of tracer is added in years 0, 1, and 2?
1. 2132 2. 1
2 3. 316 4. 9
16 5. none of above
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Divide and Conquer
The content of barrel #3 has no direct dependence on barrel #1.
The new content of barrel #3 depends only on itself and barrel #2.
All dependence on barrel #1 is through barrel #2.
1 0 0year 0
12
12 0year 1
14
12
14year 2
Since barrel #3 depends only on barrel #2, and barrel #2 depends
only on barrel #1, the three barrel system is equivalent to the cas-
cade of three one barrel systems!
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Divide and Conquer
Making a three-barrel system by cascading three one-barrel systems.
3-barrelsolera
x[n] y[n]
Year Tracer in Barrel Barrel Barrel Tracer outn x[n] = δ[n] #1 #2 #3 y[n] = h3[n]0 1 1 0 0 01 0 1/2 1/2 0 02 0 1/4 2/4 1/4 03 0 1/8 3/8 3/8 1/84 0 1/16 4/16 6/16 3/165 0 1/32 5/32 10/32 6/326 0 1/64 6/64 15/64 10/64
1
n
x[n] = δ[n]3/16
n
y[n] = h3[n]
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Divide and Conquer
Find the unit-sample response h1[n] for a 1-barrel solera.
1-barrelsolera
1-barrelsolera
1-barrelsolera
x[n] y[n]
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Divide and Conquer
Find the unit-sample response h1[n] for a 1-barrel solera.
1-barrelsolera
x[n] h1[n]
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Divide and Conquer
How can we use this to find h2[n] (output of a two-barrel system)?
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Divide and Conquer
How can we use this to find h3[n] (output of a three-barrel system)?
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