Wavenumber-Frequency Space. Material drawn from Sec. 2.5

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1 Where We Are in J&D Wavenumber-Frequency Space ECE 6279: Spatial Array Processing Spring 20 Lecture 3 Material drawn from Sec. 2.5 For now, we will skip Sec. 2.6 on random space-time fields (but we will come back to those ideas later Prof. Aaron D. Lanterman School of Electrical & Computer Engineering Georgia Institute of Technology AL: <lanterma@ece.gatech.edu> Different Definitions of FT Pairs S eng ( = s(texp( jtdt s(t = S eng (exp( jtd Mathematician s style: S math ( = s(texp(+ jtdt s(t = S math (exp( jtd Adapting an Engineer s FT Table S math ( = s(t = = s(texp(+ jtdt = s(texp[ j( t ]dt = S eng ( = F eng {S math }( t S math (exp( jtd S math (exp[ j( t ]d

2 Fourier Transforms of a Delta Function S( = δ(texp( jtdt = δ(texp( j0dt Mathematician s style: S( = δ(texp(+ jtdt = = δ(tdt = Inverse FT of a Delta Function Engineer s style (works for math style too: s(t = δ(exp( jtd = δ(exp( j0td = F δ( From previous slide: δ(t F Time Shift Property: Engineer s Style eng S timesh ( = exp( jtdt Time Shift Prop.: Mathematician s Style Mathematician s style: S math timesh ( = exp(+ jtdt substitute t = t t 0, t = t + t 0 substitute t = t t 0, t = t + t 0 eng S timesh ( = s( t exp[ j( t + t 0 ]dt = exp( jt 0 s( t exp( j t d t = exp( jt 0 S eng ( math ( = s( t exp[ + j( t + t 0 ]dt S timesh = exp(+ jt 0 s( t exp(+ j t d t = exp(+ jt 0 S math ( 2

3 Freq. Shift Property: Engineer s Style s freqsh (t = S eng ( 0 exp( jtd substitute = 0, = + 0 s freqsh (t = S eng ( exp[ j( + 0 t ]d = exp( j 0 t S eng ( exp( j td = exp( j 0 ts(t Freq. Shift Prop.: Mathematicians s Style Mathematicians s style: s freqsh (t = S math ( 0 exp( jtd substitute = 0, = + 0 s freqsh (t = S math ( exp[ j( + 0 t ]d = exp( j 0 t S math ( exp( j td = exp( j 0 ts(t Quick Proofs of Math-Style Shift Props. Time shift: Feng exp( jt 0 S eng ( Fmath exp(+ jt 0 S eng ( = exp(+ jt 0 S math ( Frequency shift: exp( j 0 ts(t Feng S eng ( + 0 exp( j 0 ts(t Fmath S eng ( + 0 = S math ( 0 Special Case: Deltas and Constants δ Feng exp( jt 0 S eng ( exp( j 0 ts(t Feng S eng ( 0 Mathematician s style: δ Fmath exp(+ jt 0 S math ( exp( j 0 ts(t Fmath S math ( 0 δ δ 3

4 Transforms of Delta Functions Engineers s style: δ(t t 0 Feng exp( jt 0 exp( j 0 t Feng δ( 0 Mathematicians style: δ(t t 0 Fmath exp(+ jt 0 exp( j 0 t Fmath δ( 0 Space-Time FT Pair A 4-D S-T Fourier transform S( k, = s( x,texp j t k x A 4-D S-T Inverse Fourier s( x,t = S( k transform,exp{ j( t k ( x }dk d 4 d d dk z { ( } dx dt dxdydz Engineer s style in time Mathematician s style in space Take Home Message Just like any -D function can be written as a weighted integral of complex exponentials exp jt any space-time signal - even nonpropagating ones! - can be written as a weighted integral of propagating plane waves ( exp j t k x ( Monochromatic Plane Wave What s the 4-D S-T FT of S( k, = { ( } s( x,t = exp j 0 t k 0 x { ( } s( x,texp j t k x dx dt = exp( j 0 texp( jtexp( jk 0 xexp( jk x dxdt = ( 4 δ( k k 0 δ( 0 δ( v where δ(v x δ(v y δ(v z A point in wavenumberfrequency space 4

5 General Plane Wave What s the 4-D S-T FT of ss( x,t = s(t α 0 x Axes for Showing S-T Fourier Support Notation borrowed from Chris Barnes Take Eng. ss( x FT in time domain, = S(exp( j first: α 0 x Then SS( k, Math. FT in spatial = S(( 3 δ( k domain: α 0 A line in wavenumber-frequency space Narrowband, Nonpropagating S-T Signal Wideband, Directional S-T Signal Not fixing a specific speed of propagation! = 0 k ζ 0 5

6 Wideband, Iso., Fixed-Speed S-T Signal Narrowband, Iso., Fixed-Speed S-T Signal Now fixing a specific speed of propagation Now fixing a specific speed of propagation = c k 0 = c k and fixing frequency Isotropic: propagating in all directions Isotropic: propagating in all directions Wideband, Dir., Fixed-Speed S-T Signal Narrowband, Dir., Fixed-Speed S-T Signal Now fixing a specific speed of propagation Now fixing a specific speed of propagation = c k 0 = c k and fixing direction and fixing frequency and fixing direction k ζ 0 6

7 Monochromatic Spherical Wave What s the 4-D S-T FT of s(r,t = exp{ j( 0 t k 0 r }/r With polar wavenumber coordinates: S(k, = 2 jk δ(k k 0 + 4π k 2 (k 0 2 δ( 0 (at least according to J&D, p. 44 exp( jt ss( x, Space-frequency Element space Post-Doppler exp( j k x Doug Williams Chart ss( x,t Space-Time exp( jt Element space Pre-Doppler exp( j k x exp( j exp( j k x k x Ss( k,t Wavenumber-time Beamspace Pre-Doppler SS( k exp( jt, exp( jt Wavenumber-frequency Beamspace Post-Doppler Filtering to Extract Information Filter data in wavenumber-frequency space: Y( k, = H( k,s( k, Ideal examples: Focus on one frequency H( k, = δ( 0 Focus in one direction H( k, = δ( k k 0 Spatiotemporal Convolution Multiplication in Fourier domain Y( k, = H( k,s( k, Corresponds to convolution in space-time domain: y( x,t = h( x ξ,t τs( ξ,τ d ξ dτ Hence ideal filters on previous slide aren t practical - have infinite extent in space-time 7

8 Spatiotemporal Filter Design Problem Challenge is to find a space-time impulse response h( x,t that gets close to the desired H( k, under some constraints: If we want real-time implementation, temporal support must be restricted to t>0 (causality Tricks from ECE4270 come into play More importantly, spatial support must be limited to where you can put sensors! New spin in ECE6279 8