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.
Convolution 1D and 2D signal processing Consider the delta function x ( n) ( n) x(0) n 0 else (t)dt Time-shift delta (n k ) 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) v(t ) (t n v s (t) v(t)s(t) n n / f s) What does sampling 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 V( f ) F[v(t )] v(t) e i F V( f ) cos v(t )e V( f )e i sin ift ift dt dt Euler’s identity e i cos i sin Sine-cos Rep x(t) n v(t) a0 an cos(2 nf t) n bn sin(2 nf t) (a1 cos t b1 sin t) (a2 cos 2t b2 sin 2t) K Harmonic Analysis a0 an bn T T T T T x(t)dt x(t)cos(2 nf t)dt T x(t )sin(2 nf t)dt 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 ) n V s ( f ) V(F) * Vs ( f ) (t fs n n n / fs) fs ( f nf s ) V(f nf s ) Spectrum reproduced Vs ( f ) fs V(f nf s ) n spectrum to be reproduced at intervals fs