Convolution: 1D and 2D signal processing

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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.

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Nội dung Text: Convolution: 1D and 2D signal processing

Convolution 1D and 2D signal processing
Consider the delta function x(0) n 0 x ( n) ( n) 0 else (t)dt 1
Time-shift delta k (n k ) x k x(k ) k (t td )
Sample the input (it’s a convolution!) ( * x)[n] [n k ]x[k ] x[n] k s(t) (t n / fs) 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 cos t b1 sin t) (a2 cos 2t b2 sin 2t) K
Fourier Coefficients a0 ,a1 ,b1 , a2 ,b2 K v(t) a0 (a1 cos t b1 sin t) (a2 cos 2t b2 sin 2t) K
CTFT 2 ift V( f ) F[v(t )] v(t )e dt 1 2 ift v(t) F V( f ) V( f )e dt i e cos i sin
Euler’s identity i e cos i sin
Sine-cos Rep x(t) an cos(2 nf 0 t) bn sin(2 nf 0 t) n 0 n 1 v(t) a0 (a1 cos t b1 sin t) (a2 cos 2t b2 sin 2t) K
Harmonic Analysis 1 T a0 x(t)dt T 0 2 T an x(t)cos(2 nf 0 t)dt T 0 2 T bn x(t )sin(2 nf 0 t)dt T 0
Convolution=time-shift&multi 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
Sample v s (t) v(t)s(t) v(t ) (t n / fs) n V s ( f ) V(F) * fs ( f nf s ) n Vs ( f ) fs V(f nf s ) n
Spectrum reproduced Vs ( f ) fs V(f nf s ) n spectrum to be reproduced at intervals fs