Convolution: 1D and 2D signal processing

Convolution: 1D and 2D signal processing provides about consider the delta function; time-shift delta; sample the input; Fourier Coefficients; Euler’s identity; Sine-cos Rep; Harmonic Analysis; Convolution Thm; Spectrum reproduced.

1D and 2D signal processing

Consider the delta function

      else       

0

0   

) 0

( )

( )

x

(t)dt 1

Time-shift delta

)

(n k

k

k

x ( )

Sample the input (it’s a

convolution!)

] [ ]

[ ] [

] )[

*

k

s(t) (t n / f s)

n

v s (t) v(t)s(t) v(t ) (t n / f s )

n

What does sampling do to

spectrum?

What is the spectrum?

v(t) a0 (a1 cost b1sin t) (a2 cos2t b2sin 2t) K

Fourier Coefficients

v(t) a0 (a1 cost b1sin t) (a2 cos2t b2sin 2t) K

CTFT

Euler’s identity

Sine-cos Rep

n 1

n 0

v(t) a0 (a1 cost b1sin t) (a2 cos2t b2sin 2t) K

Harmonic Analysis

T

T

T

V * W( f ) V ( )W( f )d

Convolution Thm

V * W( f ) F(v(t)w(t))

multiplication in the time domain =

convolution in the frequency domain

n

n

n

Spectrum reproduced

n

spectrum to be reproduced at intervals

f s