Convolution

1D and 2D signal processing

Consider the delta function

⎩⎨⎧ =

=⋅ else 0

0 )0()()(

nxnnx δ

δ(t)− ε

ε

∫ dt = 1

Time-shift delta

)( knk −=δδ

kk kxx δδ ⋅=⋅ )(δ( t − td )

Sample the input (it’s a convolution!)

][][][])[*( nxkxknnxk∑∞

−∞=

=−= δδ

s(t) = δ(t −n / fs)n=−∞

∞

∑

vs (t) =v(t)s(t) =v(t) δ(t −n/ fs)n=−∞

∞

∑

What does sampling do to spectrum?

What is the spectrum? v(t) =a0 + (a1 cost+b1sint) + (a2 cos2t+b2sin2t)+K

Fourier Coefficients

a0 ,a1,b1, a2 ,b2K

v(t) =a0 + (a1 cost+b1sint) + (a2 cos2t+b2sin2t)+K

CTFT

V( f ) =F [v(t)] = v(t)e−2 πiftdt−∞

∞

∫

v(t) =F−1 V( f )[ ] = V( f)e2πiftdt−∞

∞

∫e iθ =cosθ + i sinθ

Euler’s identity

e iθ =cosθ + i sinθ

Sine-cos Rep

x(t) = an cos(2πnf0t) + bn sin(2πnf0t)n=1

∞

∑n=0

∞

∑

v(t) =a0 + (a1 cost+b1sint) + (a2 cos2t+b2sin2t)+K

Harmonic Analysisa0 =1

Tx(t)dt

0

T

∫

an =2T

x(t)cos(2πnf0t)dt0

T

∫

bn =2T

x(t)sin(2πnf0t)dt0

T

∫

Convolution=time-shift&multi

V *W( f ) ≡ V(λ )W( f −λ)dλ−∞

∞

∫

Convolution ThmV *W( f ) =F(v(t)w(t))

multiplication in the time domain =convolution in the frequency domain

Sample

vs (t) =v(t)s(t) =v(t) δ(t −n/ fs)n=−∞

∞

∑

Vs ( f ) =V(F) * fsδ( f −nfs )n=−∞

∞

∑

Vs ( f ) = fs V( f −nfs)n=−∞

∞

∑

Spectrum reproduced

Vs ( f ) = fs V( f −nfs)n=−∞

∞

∑

spectrum to be reproduced at intervalsf s