EE 355 / GP 265 Homework 3 Solutions Winter

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1 EE 355 / GP 265 Homework 3 Solutions Winter February 8, Doppler centroid (a) Plot of the azimuth spectrum of the data averaged over all valid range bins: 90 Problem 1a: average azimuth spectrum Magnitude (db) Frequency (Hz) First, create the reference chirp signal for ERS parameters, then correlate each range line with it to get the range-compressed image. Then keep only the valid data from the correlation. (The original ERS data in complex form has 4903 range bins, and the reference chirp has 703 samples, so I keep =4200 range bins in the range-compressed image.) Then compute the FFT of each azimuth line in the range-compressed image, and average the azimuth spectrum over the 4200 valid range bins. 1

2 (b) Determine the Doppler centroid using the spectrum in (a) and also the average phase change method. Using the spectrum in (a), the Doppler centroid frequency is at the maximum value, except for the anomalous spike at 0 Hz: -300 Hz. For the average phase change method, compute the phase shift ϕ(ir) at each range bin ir from line iaz to line iaz 1 in azimuth (here R is the range-compressed image): P (ir) = Naz iaz=2 R(ir, iaz)r 1 Im(P (ir)) (ir, iaz 1), ϕ(ir) = tan Re(P (ir)) (1) Average ϕ(ir) over all range bins to compute the Doppler centroid f D = Hz. ϕ = Nr ir=1 ϕ(ir), Nr ϕ 2π = f D prf f D = prf ϕ 2π (2) function c h i r p=makechirp ( s, tau, fs, fc, s t a r t, n) %s : s l o p e %tau : p u l s e l e n g t h %f s : sample r a t e %f c : c e n t e r frequency %s t a r t : s t a r t i n g l o c a t i o n of c h i r p %n : number o f samples dt=1/ f s ; % sampling time i n t e r v a l npts=floor ( tau / dt ) ; % number of p o i n t s of the c h i r p s i g n a l t=( npts / 2 : npts /2 1) dt ; phase=pi s t.ˆ2+2 pi f c t ; % c h i r p =[ z e r o s (1, s t a r t 1) exp (1 i phase ) z e r o s (1,n l e n g t h ( phase) s t a r t +1)] c h i r p=zeros (n, 1 ) ; c h i r p ( s t a r t : s t a r t+npts 1)=exp(1 i phase ) ; return % Unfocused SAR p r o c e s s o r p a r t 1 : range compression & determine d o p p l e r c e clear ; clc ; close a l l ; %% do we need to do range compression? i f so, s e t f l a g r g c p =1 f l a g r g c p =1; n l i n e =10100; nrgbin =( )/2; %% make c h i r p 2

3 s = e11 ; % Chirp slope, Hz/ s tau =37.12 e 6; % Pulse length, s f s =18.96 e6 ; % Complex sample rate, Hz f c =0; % Center frequency ( Carrier frequency ) Nsample =( )/2; % Number of samples N=Nsample ; % c h i r p l e n g t h dt=1/ f s ; % Sample spacing in time domain df=f s /N; % Sample spacing in frequency domain f=( N/ 2 :N/2 1) df ; r e f c h i r p=makechirp ( s, tau, fs, fc, 1, Nsample ) ; r e f s p e c t =( f f t ( r e f c h i r p ) ) ; r e f s p e c t d b =20 log10 ( abs ( r e f s p e c t )+1e 30); i f f l a g r g c p==1 %% read data f i d=fopen ( e r s d a t a. hw3, r ) ; data=fread ( f i d, [ , ], uint8 ) ; data=data ; fclose ( f i d ) ; data=data 15.5; n l i n e =10100; data1=data ( :, : ) ; r e a l d=data1 ( :, 1 : 2 : ) ; imagd=data1 ( :, 2 : 2 : ) ; datacp=complex ( reald, imagd ) ; %% range compression s p e c t d a t a=f f t ( datacp, Nsample, 2 ) ; data rgcp=zeros ( n line, nrgbin ) ; for i =1: n l i n e data rgcp ( i, : ) = ( i f f t ( s p e c t d a t a ( i, : ). conj ( r e f s p e c t. ) ) ) ; figure imagesc ( abs ( data rgcp ) ) colormap( gray ) t i t l e ( Range compressed image ) xlabel ( range ) ylabel ( azimuth ) axis on 3

4 save ( d a t r c, data rgcp ) %% Average azimuth s p e c t r a load d a t r c % azimuth parameters p r f =1679.9; % Pulse r e p e t i t i o n frequency, Hz r 0 =830000; % Near range, m v e l =7550; % Platform v e l o c i t y, m/ s wvlen =0.0566; % Wavelength, m dt2=1/ p r f ; df2=p r f / n l i n e ; n v a l i d=nsample floor ( tau f s ) ; rcomp=data rgcp ( :, 1 : n v a l i d ) ; s p e c t a z=f f t ( rcomp ) ; s p e c t a v e r a z=mean( abs ( s p e c t a z ), 2 ) ; spectdb az =20 log10 ( abs ( s p e c t a v e r a z ) ) ; f 2=linspace( p r f /2, p r f /2, n l i n e ) ; figure plot ( f2, f f t s h i f t ( spectdb az ) ) xlabel ( f ) ylabel ( S ( f ), db ) t i t l e ( Average Azimuth Spectrum ) grid on ; saveas ( gcf, pb1 a. t i f, t i f f ) % e s t i m a t e f d c e n t r o i d using the average phase change method for i =1: n v a l i d bin=rcomp ( 2 :, i ). conj ( rcomp ( 1 : 1, i ) ) ; r e s u l t=sum( bin ) ; phase ( i )=atan (imag( r e s u l t )/ real ( r e s u l t ) ) ; avgphase=mean( phase ) ; fdcen=avgphase /2/ pi p r f ; d i s p l a y ( fdcen, Doppler c e n t r o i d i s estimated to be ) save ( fdcen, fdcen ) 2. An unfocused SAR processor 4

5 (a) Parameters of an unfocused processor for the radar in (1). The azimuth resolution is: δ az = λr 0 = ( m)( m) = m (3) The pulse spacing in azimuth is: p = v prf 7550 m/s = = m (4) Hz The minimum number of pulses is (using Method 2 in Handout 19): δaz m min np = = = 49 pulses (5) p m Rounding up to the next highest power of 2, we use np = 64 pulses in a burst. The frequency resolution is δf = prf np The output pixel spacing is then = Hz 64 pulses = Hz (6) δx = (δf)λr 0 2v = (26.25 Hz)( m)( m) (2)(7550 m/s) = m (7) The burst length is δt = np prf The repeat cycle time for a 10 m long antenna is: 64 pulses = = s (8) Hz t cycle = r 0λ lv = ( m)( m) (10 m)(7550 m/s) = s (9) (b) Single-patch unfocused image from the first np = 64 lines of the file (with 4 range looks). The effect of the antenna pattern is clearly visible. The targets in the middle of the patch are brighter than the ones on the edges. To take range looks, we downsample in the range direction by a factor of 4 so that the output pixel spacing in the range direction is equal to the output pixel spacing in the azimuth direction (δx = m), which will give us square pixels. 5

Azimuth samples 20 40 60 Problem 2b: Single patch unfocused image, 4 range looks 100 200 300 400 500 600 700 800 900 1000 Range samples 10 4 8 6 4 2 (c) Patch to patch spacing (offset between

6 Azimuth samples Problem 2b: Single patch unfocused image, 4 range looks Range samples (c) Patch to patch spacing (offset between patches) if we want to process all of the data: patch = ( p)(np) δx = (4.494 m)(64 pulses) (81.66 m) = pixels (10) The patch spacing is not a whole number, so we alternate between taking 3 and 4 pixels for the patch-to-patch spacing when computing the multi-looked image in part (d). (d) Image from multi-looked unfocused processor. First compute the number of looks (patches) given the amount of data: naz az lines nlooks = = = 157 looks (11) np 64 pulses Then compute the number of azimuth pixels in the multi-looked image, to preallocate space: N az,look = np + (nlooks 1) patch = 613 samples (12) Multi-look unfocused image, without taking range looks: Azimuth samples Problem 2d: Multi-look unfocused image Range samples

7 Multi-look unfocused image, with 4 range looks (so we have square pixels): this looks like a map of the San Francisco Bay Area. Azimuth samples Problem 2d: Multi-look unfocused image, 4 range looks Range samples % Unfocused SAR p r o c e s s o r part2 : % Range d o p p l e r p r o c e s s i n g clear ; clc ; close a l l % c h i r p parameter s = e11 ; % Chirp slope, Hz/ s tau =37.12 e 6; % Pulse length, s f s =18.96 e6 ; % Complex sample rate, Hz f c =0; % Center frequency ( Carrier frequency ) Nsample =( )/2; % Number of samples % more parameters load d a t r c load fdcen p r f =1679.9; % Pulse r e p e t i t i o n frequency, Hz r 0 =830000; % Near range, m v e l =7550; % Platform v e l o c i t y, m/ s wvlen =0.0566; % Wavelength, m l =10; % Antenna l e n g t h 7

8 n v a l i d=nsample floor ( tau f s ) ; % v a l i d number of range of b i n s rcomp=data rgcp ( :, 1 : n v a l i d ) ; Naz =10100; % range d o p p l e r mapping parameters d e l t a a z =(wvlen r0 ) ˆ 0. 5 ; % azimuth r e s o l u t i o n pulsp=v e l / p r f ; % p u l s e spacing Nmin=round( d e l t a a z / pulsp ) ; p a t c h s i z e =2ˆ c e i l ( log (Nmin)/ log ( 2 ) ) ; l e n b u r s t=p a t c h s i z e / p r f ; % b u r s t l e n g t h d e l t a f r e q=p r f / p a t c h s i z e ; % f r e q u e c y r e s o l u t i o n d e l t a x= d e l t a f r e q wvlen r0 /2/ v e l ; % output p i x e l spacing azbw= r0 wvlen / l ; % Azimuth beamwidth i n t e r v a l=azbw/ v e l ; % r e p e a t i n t e r v a l %% Doppler c e n t r o i d c o r r e c t i o n rcompfd=zeros (Naz, n v a l i d ) ; for i =1:Naz az=rcomp ( i, : ) exp( (1 j ) 2 pi fdcen i / p r f ) ; rcompfd ( i, : ) = az ; % check i f we have done doppler c e n t r o i d c o r r e c t i o n c o r r e c t l y s p e c t a z=f f t s h i f t ( f f t ( rcompfd, [ ], 1 ) ) ; s p e c t a v e r a z=mean( abs ( s p e c t a z ), 2 ) ; spectdb az =20 log10 ( abs ( s p e c t a v e r a z ) ) ; figure plot ( spectdb az ) xlabel ( f ) ylabel ( S ( f ), db ) t i t l e ( Average Azimuth Spectrum a f t e r Doppler c e n t r o i d c o r r e c t i o n ) grid on ; %% p r o c e s s the f i r s t patch n l i n e=p a t c h s i z e ; d a t a s i n g l e=rcompfd ( 1 : nline, : ) ; % azimuth transform d a t a s i n g l e f o u r=abs ( f f t s h i f t ( f f t ( d a t a s i n g l e, [ ], 1 ), 1 ) ) ; % average range l o o k s nrlook =4; 8

9 imagetest=zeros ( n line, floor ( n v a l i d / 4 ) ) ; for i =1: floor ( n v a l i d /4) imagetest ( :, i )=sum( d a t a s i n g l e f o u r ( :, ( i 1) nrlook +1: i nrlook ), 2 ) ; figure imagesc ( imagetest ) colormap( gray ) t i t l e ( Unfocused SAR image o f the f i r s t patch, 4 range l o o k s ) xlabel ( range ) ylabel ( azimuth ) cax=caxis ; caxis ( 0. 5 cax ) axis image saveas ( gcf, p b 2 s i n g l e p a t c h. eps, eps ) %% p r o c e s s a l l patches npatch=floor ( Naz/ p a t c h s i z e ) ; pix=pulsp p a t c h s i z e / d e l t a x ; n a z t o t a l=round( p a t c h s i z e +(npatch 1) pix ) ; % range compression a l r e a d y done! d a t a a l l=rcompfd ; % azimuth transform d a t a a l l f o u r=zeros ( naztotal, n v a l i d ) ; d a t a a l l f o u r ( 1 : patchsize, : ) = d a t a s i n g l e f o u r ; for i =2: npatch d i s p l a y ( i, p r o c e s s i n g the i t h patch ) d a t a s i n g l e=d a t a a l l ( ( i 1) p a t c h s i z e +1: i patchsize, : ) ; d a t a s i n g l e f o u r=abs ( f f t s h i f t ( f f t ( d a t a s i n g l e ), 1 ) ) ; o f f s e t=round ( ( i 1) pix ) ; d a t a a l l f o u r ( o f f s e t : o f f s e t+patchsize 1, : ) =... ( d a t a s i n g l e f o u r ( 1 : patchsize, : ) +... d a t a a l l f o u r ( o f f s e t : o f f s e t+patchsize 1, : ) ) ; %average range l o o k s nrlook =4; i m a g e f i n a l=zeros ( naztotal, floor ( n v a l i d / 4 ) ) ; for i =1: floor ( n v a l i d /4) i m a g e f i n a l ( :, i )=sum( d a t a a l l f o u r ( :, ( i 1) nrlook +1: i nrlook ), 2 ) ; 9

10 figure imagesc ( i m a g e f i n a l ) colormap( gray ) t i t l e ( Multi look unfocused image, 4 range l o o k s ) xlabel ( range ) ylabel ( azimuth ) set ( gca, YDir, normal ) cax=caxis ; caxis ( 0. 4 cax ) axis image saveas ( gcf, pb2. eps, eps ) 10

EE 355 / GP 265 Homework 5 Solutions

EE 355 / GP 265 Homework 5 Solutions EE / GP Homework Solutions February,. Autofocus: subaperture shift algorithm Compress the signal (with chirp slope s = Hz/s) by correlating it with the reference chirp (with different chirp slope s ) twice: