Continuous-Time Convolution

Linear Systems and SignalsLecture 5

Spring 2008

5 - 2

Convolution Demos• Johns Hopkins University Demonstrations

http://www.jhu.edu/~signals

Convolution applet to animate convolution of simple signals and hand-sketched signals

Convolve two rectangular pulses of same width gives a triangle (see handout E)

• Some conclusions from the animationsConvolution of two causal signals gives a causal result

Non-zero duration (called extent) of convolution is the sum of extents of the two signals being convolved

5 - 3

Transmit One Bit• Transmission over communication channel (e.g.

telephone line) is analog

h t

)(th

1

p t

)(1 tx

A

‘1’ bit

t

p

)(0 tx

-A

‘0’ bit

Model channel as LTI system with impulse response

h(t)

CommunicationChannel

input output

x(t) y(t)t

t

)(1 ty receive‘1’ bit

)(0 ty

-A Th

receive ‘0’ bit

h+p

th+p

h

h

Assume that Th < Tp

A Th

5 - 4

Transmit Two Bits (Interference)• Transmitting two bits (pulses) back-to-back

will cause overlap (interference) at the receiver

• How do we prevent intersymbolinterference at the receiver?

h t

)(th

1

Assume that Th < Tp

tp

)(tx

A

‘1’ bit ‘0’ bit

p

* =)(ty

-A Th

tp

‘1’ bit ‘0’ bit

h+p

intersymbol interference

5 - 5

Transmit Two Bits (No Interference)• Prevent intersymbol interference by waiting Th

seconds between pulses (called a guard period)

• Disadvantages?

h t

)(th

1

Assume that Th < Tp

* =

tp

)(tx

A

‘1’ bit ‘0’ bit

h+p

t

)(ty

-A Th

p

‘1’ bit ‘0’ bit

h+p

h

5 - 6

m

mnxmhny ][ ][ ][ dtxhty

h[n] y[n]x[n]

LTI systemrepresentedby its impulseresponse

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

LTI system representedby its impulseresponse

Discrete-time Convolution Preview• Discrete-time

convolution

• For every value of n, we compute a new summation

• Continuous-time convolution

• For every value of t, we compute a new integral

5 - 7

1

0

][ ][ ][N

m

mnxmhny

z-1 z-1 z-1…

…

x[n]

y[n]

h[0] h[1] h[2] h[N-1]

Discrete-time Convolution Preview• Assuming that h[n] has finite

duration from n = 0, …, N-1• Block diagram of an implementation (finite

impulse response digital filter): see slide 2-4

5 - 8

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