A radar receiver records an echo as complex samples and locates the returned pulse by…

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Computational Physics · Scientific programming for physicists

Scientific Computing for Physicists, Ch Spectra with error bars: the fast transform, convolution and spectral estimation

Problem

A radar receiver records an echo as

N=1024

complex samples

x_n

and locates the returned pulse by cross-correlating the echo with the transmitted chirp,

M=255

complex samples

y_p

c_m=\sum_px_{p+m}\,y_p^{*},

summed over every

p

for which both samples exist, for each lag

m

from

-(M-1)

N-1

. The correlation is computed by zero-padding both records to a length

L

, multiplying the echo's DFT

X_k

by the complex conjugate

Y_k^{*}

of the chirp's DFT, and inverse transforming. ((a)) Starting from the orthogonality relation,

\frac{1}{L}\sum_{k=0}^{L-1}e^{2\pi ikj/L}

equal to

1

when

j

is a multiple of

L

and

0

otherwise, prove that the inverse transform of

X_kY_k^{*}

is the circular cross-correlation

z_n=\sum_{p=0}^{L-1}x_{(p+n)\bmod L}\,y_p^{*}

of the two zero-padded sequences. ((b)) Show that, for general records,

z_n=c_n

for

0\le n\le N-1

and

z_{L+m}=c_m

for

-(M-1)\le m\le-1

exactly when

L\ge N+M-1

. Give that smallest

L

and the smallest power of two no less than it. ((c)) A programmer skips the extra padding and uses

L=N=1024

. Find which output indices

n

are corrupted, and which lags have been wrapped onto them. In one echo the pulse starts at sample

600

, so the correlation peaks at lag

600

: is that peak read correctly?