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- Convolution

(Last revised: 2/13/2015)

EE110, S15: Circuits and Systems, Lecture 5

Prof. Ping Hsu

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Before we introduce the mathematical definition of convolution integral and its relationship with transfer function, lets first consider a linear time invariant (LTI) discrete time system.

Discrete time LTI system

X(k) Y(k)

The system is a dynamical system; namely, it has a memory effect. Its output depends on the current input as well as all past inputs. Consider the following input and output strings.

From inspection, you can probably guess that the output is the sum of the current input and half of the previous input. This half of the previous input characterizes the systems memory.

Time 1 2 3 4 5 6 7 8 9X(k) 4 2 0 -6 -2 4 4 2 0Y(k) -- 4 1 -6 -5 3 6 4 1

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Consider the following input and output strings.

Time 1 2 3 4 5 6 7 8 9 10 11 12 13X(k) 4 2 0 -6 -2 4 4 2 -10 6 2 -2 6Y(k) -4.6 4.4 4.8 2.2 -7.6 -0.6 8.6 -5.2 5.2

From these input and output data string, it is not easy to see the memory characteristic of this system. The output from the following string of input (called pulse function) reveals the memory effect of the system clearly.

Time 1 2 3 4 5 6 7 8 9 10 11 12 13X(k) 0 0 0 0 1 0 0 0 0 0 0 0 0Y(k) -- 0.8 0.5 -0.4 0.2 -0.1 0 0 0 0

From this output, it is easy to see that the memory effect can be described as:The current input is weighted at 0.8, the previous input is weighted at 0.5,the input before that is weighted at -0.4, before that is 0.2, before that is -0.1, and all inputs before that are completely forgotten.

This string of values is called pulse response of the system.

-0.1 0.2 -0.4 0.5 0.8

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Time 1 2 3 4 5 6 7 8 9 10 11 12 13X(k) 4 2 0 -6 -2 4 4 2 -10 6 2 -2 6

To find the output value at a certain time instance, we just need to flip the pulse response and right- align it to the input at the time instance that we want to find the output value. Two examples are give below: one for t=6 second and one for t=13 second,

Y(k) -4.6 4.4 4.8 2.2 -7.6 -0.6 8.6 -5.2 5.2-0.1 0.2 -0.4 0.5 0.8

Conclusion:

The pulse response of a discrete time system completely characterizes the input-output relationship of a LTI system. The time flipped version of the pulse response is the systems forgetting curve. The output of the system can be determined by sliding the forgetting curve from left to right. At each time instance, the output is the sum of the current and all the previous inputs weighted by the forgetting curve.

(4)(0.8)+(-2)(0.5)+(-6)(-0.4)+(0)(0.2)+(2)(-0.1)=4.4

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Discrete time LTI system

Any input

Pulse responsepulse

Flipped pulse response(forgetting curve)

Output

The output at any time instance is the sum of present and all prior inputs weighted by the time flipped version of the pulse response (forgetting curve).

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Continuous time LTI system

Any input

Impulse response h(t)Impulse (t)

Flipped and shifted impulse response

Output

The output at any time instance (to) is the integral of the product of the input and the time flipped version of the impulse response.

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to

h(to-t)

x(t)

y(t)

0

0( ) ( ) ( )t

oy t x t h t t dt

y(t0)

( ) ( ) ( )t

y t x h t d or This is the convolution integral of x(t) and h(t),

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8- Convolution Property

where

is called convolution integral

H(s)

( )* ( ) ( ) ( )x t h t H s X s

X(s) Y(s) = H(s)X(s)

h(t)x(t) ( ) ( ) ( )t

y t x h t d Freq. Domain:

Time Domain:

( )* ( ) ( ) ( )t

x t h t x h t d

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- The response of a system to an impulse ((t)) input is called the impulse response (h(t)) of the system.

- For Linear & Time-Invariant (LTI) systems, if all initial conditions are zero, the output y(t) can be computed by taking the convolution integral of the input x(t) and the impulse response h(t) , i.e.,

x(t) (t) y(t) h(t)LTI

0( ) ( )* ( ) ( ) ( )

ty t x t h t x h t d

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Systematic Steps to Evaluate the Convolution Integral

Below

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Below, you can exchange x & h and still get the same y(t)

2013 National Technology and Science Press. All rights reserved.

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Find vout(t) using the convolution integral.

/ 21( ) ( ) 2 ( )t th t e u t e u t

( ) ( )* ( ) ( ) ( )out in inv t v t h t v t h t d

Example: RC Circuit Response to Rectangular Pulse

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For t

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1 2( )

0

1 2( ) 2 2(1) 0 2 2

0

( ) 2

12 2 12

tout

t t t t

v t e d

e e d e e e e e

For 1 twhen t=2

Example: RC Circuit Response to Rectangular Pulse14

2013 National Technology and Science Press. All rights reserved.

The output is:

2

2 2

0, 0( ) 1 ), 0 1

11 ,

(outt

tv t e t

te e

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Example

For x(t) = triang(t)u(t) shown and h(t) =u(t), calculate and sketch y(t) = x(t)*h(t)

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1

x()

0

1h(-)

0

Step 1: Sketch x() and h(-)a) t < 0,

Step 2 : sketch h(t-)

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1

x(t)

0t

1h(t)

0 t

1h(t-)

0 x()h(t-)

0

t

Step 3: Sketch the product curve x()h(t-) Here, for t < 0, x()h(t-) = 0 for all , hence the area under it = 0. y(t) = x(t)*h(t) = 0, t < 0 . (1)

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Step 4b) 0 < t < 1,

Step 2sketch h(t-) and Step 3Sketch the product curve x()h(t-) & calculate the area under it

y(t) t 1 (1 t)2

t (1 12

t), 0 t 1, ... (2)

h(t-)

0

t

1

t

1

1

x()

0

1

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1-

x()h(t-) area =y(t)1-t

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1

x()

0 1

t

h(t-)

0

1

0

1-

1

x()h(t-) area =y(t)1-

Step 4c) t > 1,

Step 2sketch h(t-) and Step 3Sketch the product curve x()h(t-) & calculate the area under it

y(t) 121 t 1

2, t 1, ... (3)

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We now put the pieces together:

y(t) x(t)*h(t) x( )h(t )d

0, t 0 t(1 1

2t), 0 t 1

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, t 1Note that the signal y(t) covers the full range t .

1/2

y(t)

0 t1

t(1 12

t)

Example (cont.)