EE104: Lecture 8 Outline

Review of Last Lecture

Convolution Review

Signal Bandwidth

Dirac Delta Function and its Properties

Filter Impulse and Frequency Response

Review of Last Lecture

Time ScalingDualityFrequency Shifting (Modulation)Multiplication ConvolutionConvolution Multiplication

Filter analysis often easier in frequency domain

x(t) h(t) y(t)=h(t)*x(t)

X(f) H(f) Y(f)=H(f)X(f)

LTI Filter

z(2-)

2

Convolution Reviewy(t)=x(t)*z(t)= x()z(t-)d

Flip one signal and drag it across the otherArea under product at drag offset t is y(t).

t t+1t-1

z(t-)

0 1-1

x(t)

0 1-1

z(t)

t t

0 1-1

x()

-6

z(-6-)

0 1-1

y(t)

-2 2

0-2 2 t

z(-2-)z(-1.99-)

.01

z(0-)

2

z(1-)

2

-6

-4

z(-4-)

-4

z(-1-)

1

z()

x()

Signal Bandwidth

For bandlimited signals, bandwidth B defined as range of positive frequencies for which |X(f)|>0.

In practice, all signals time-limitedNot bandlimitedNeed alternate bandwidth

definition|X(f)|

2B

0

Bandlimited|X(f)|

2B

0

Null-to-Null|X(f)|

2B

0

3dB

-3dB

Dirac Delta Function

Defined by two equations(t)=0, t=0

(t)dt=1

Alternatively defined as a limit(t)=lim0 (1/)rect(t/)

0

(t)

0

(t)

Delta Function Properties

x(t)*(t)=x(t)

(t)1

DC signals are functions in frequency.

Filter Response

Impulse Response (Time Domain)Filter output in response to a delta

input

Frequency Response (Freq. Domain)Fourier transform of impulse responseThe response of a filter to an

exponential input the same exponential weighted by H(f0)

h(t)(t) y(t)=h(t)*(t)=h(t)

H(f)

LTI Filter

Y(f)=H(f)1=H(f)

Y(f)=H(f0) ej2f0tej2f0t

Main Points

Convolution is a drag (and a flip)

Signal bandwidth definition depends on its use

Dirac delta function is a mathematical construct that is useful in analyzing signals and filters

Filter impulse response defined as filter output to delta input

Filter frequency response is Fourier transform of its impulse response