"Tobin Fricke" <fricke@ocf.berkeley.edu> wrote in message
news:Pine.SOL.4.62.0504061211410.28681@apocalypse.OCF.Berkeley.EDU...
>I am interested in learning how to do beamforming using quadrature
> sampling and/or frequency-domain processing.
>
> A few years ago I implemented a small beamformer that operated completely
> in the time domain. This was for seismic array processing; the beamformer
> combined signals for spatially separated sensors. The 'best beam' was
> found by simply trying all wavevectors on a two-dimensional grid,
> computing the resulting beam (delay-and-summed) signal, computing the
> power of each of these signals, and plotting the resulting signals on the
> wavevector grid. Seismologists sometimes call the whole process "slant
> stacking". This whole process was repeated over successive windows of the
> timeseries.
>
> I understand that there is probably a much better way to do this. I'm
> told that I can represent my signals in a "complex baseband"
> representation by doing quadrature sampling, making a complex signal with
> I and Q components, and then I will be able to perform the "delay"
> operation by multiplying by an appropriate phasor. I'm not sure this is
> exactly right. Another detail is that I would have to generate the
> quadrature signals from the pre-existing time series, maybe by calling the
> even-numbered samples "I" and the odd-numbered samples "Q"?
>
> I want to generate the 'picture' of beam power versus wave vector -- is
> there a better way to do this other than selecting wave vectors and
> computing the resulting beam power, and plotting the results? i.e. I
> imagine there might be a way to do this in one powerful step in the
> frequency domain. Here's an example of the sort of 'picture' I have in
> mind, although it's not the best plot:
>
> http://splorg.org:8080/people/tobin/projects/alaska/work/tap_output.gif
>
Tobin,
I think of them as two distinct things. Among other things, I and Q
processing is useful if the center frequency is high and the bandwidth is
relatively low or a lower percentage of the center frequency. There are
other reasons but I think this is what you meant.
Rick Lyon's book addresses the sampling method you describe.
Frequency domain beamforming might be useful if you're going to do
beamforming and spectrum analysis anyway. It may be particularly useful if
you only have access to data samples from array elements and no pre-formed
(analog) beams. From
http://www.ece.utexas.edu/~allen/EE381K-15280-F98/continuous.html
Frequency-Domain Beamforming
Beamforming can also be performed in the frequency domain, where delaying
the signal ri(t) by ti is implemented by multiplying Ri(w) by the phasor
exp(-jwti). However, we do not have access to the entire spectrum Ri(w).
Instead, we transform a segment of ri(t) by applying a finite extent window
and computing the short-time Fourier transform Ri(t,w), an approximation to
Ri(w). Now the frequency-domain beamformer output is:
Note that the frequency-domain beamformer is inherently narrowband. The
exp(-jwti) term performs the time delay, and the exp(jwt) term extracts the
component of the beamformer output b(t) at the frequency w. An approximation
to b(t) could be formed by integrating fd(t,w) over frequency.
...............................................
Either way, you end up with a map of frequency, angle data which would have
to be searched for maxima, etc. But maybe there's an efficient way to do
that. Rune probably knows all about this.
Fred
begin 666 freqDomain.gif
M1TE&.#EAG0$Z`/<``/______S/__F?__9O__,___`/_,___,S/_,F?_,9O_,
M,__,`/^9__^9S/^9F?^99O^9,_^9`/]F__]FS/]FF?]F9O]F,_]F`/\S__\S
MS/\SF?\S9O\S,_\S`/\`__\`S/\`F?\`9O\`,_\``,S__\S_S,S_F<S_9LS_
M,\S_`,S,_\S,S,S,F<S,9LS,,\S,`,R9_\R9S,R9F<R99LR9,\R9`,QF_\QF
MS,QFF<QF9LQF,\QF`,PS_\PSS,PSF<PS9LPS,\PS`,P`_\P`S,P`F<P`9LP`
M,\P``)G__YG_S)G_F9G_9IG_,YG_`)G,_YG,S)G,F9G,9IG,,YG,`)F9_YF9
MS)F9F9F99IF9,YF9`)EF_YEFS)EFF9EF9IEF,YEF`)DS_YDSS)DSF9DS9IDS
M,YDS`)D`_YD`S)D`F9D`9ID`,YD``&;__V;_S&;_F6;_9F;_,V;_`&;,_V;,
MS&;,F6;,9F;,,V;,`&:9_V:9S&:9F6:99F:9,V:9`&9F_V9FS&9FF69F9F9F
M,V9F`&8S_V8SS&8SF68S9F8S,V8S`&8`_V8`S&8`F68`9F8`,V8``#/__S/_
MS#/_F3/_9C/_,S/_`#/,_S/,S#/,F3/,9C/,,S/,`#.9_S.9S#.9F3.99C.9
M,S.9`#-F_S-FS#-FF3-F9C-F,S-F`#,S_S,SS#,SF3,S9C,S,S,S`#,`_S,`
MS#,`F3,`9C,`,S,```#__P#_S #_F0#_9@#_,P#_``#,_P#,S #,F0#,9@#,
M,P#,``"9_P"9S "9F0"99@"9,P"9``!F_P!FS !FF0!F9@!F,P!F```S_P`S
MS `SF0`S9@`S,P`S````_P``S ``F0``9@``,P``````````````````````
M````````````````````````````````````````````````````````````
M````````````````````````````````````````````````````````````
M`````````````````````"'Y! `"````+ ````"=`3H`0 C_``$('$BPH,&#
M"!,J7,BPH4.!_R(:C$C1H<2'&#-JW,BQH\>/($.*'$FRI$F0_PA63/GPXLF7
M,&/*G$FSILV;+RGJW,G3)<Z?0(,*':K1YT*C`%@61$I2J5">1%4Z73HU856F
M(ZOBQ%KSJM:87[E"]'K0)5>Q8Z<:K4AU*-NL+./&;2M5ZM>D3K&BQ:NV+]FQ
M%N\R-7L7H<^W@.TFGIBW)U3&?0U'EKEW8.&6ETLZWJR39F6%@S-##/EYX][2
MBU&*ADN7<=370=>NADV[MNW;N&&CMFHYM^_?P(,+SXEW*<;=PY,K7\X\JE;4
MR)M+G_Z[,^C9U*];-YR]>U?LWJ^7__T[6G+#M]L1][;*67W'[1QE3Q0__KQY
MANCS/V]=/NUZ_JGM)IM2Z1UV7U&*6>0:?0`69UE8HB%V&%+1J989> GZAQ^$
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MN;C^ZFKD9ZN"F[VK`HO[MIS&?I_L[^(['_R*Q"3+V8QPT[(>@%0DO>ME:X&]
MFYUQ]E<\BNF*@10L%NF8Y[P(-L]_I^*@!.=5H6CE2$(;JUS54 ?!VNGN<DD9
MX/_:Y[YQ-<N!P/M8ZH87J4"=93T>G%8`G5=7PZL9#F9<6Q42N0=$\@@1@ .S
MH5]L:$%N]4>$5UP8@3XXP[RU1"0&;!<)TZ>H,G(--& \6K6&UK3(P5" K.%<
MQ,XTQLQIQDH9'-W"*+/&GL718SH$8=L89\8GW:Q1U"&BR9872 N=D(R#5*,B
MT_;&2%JR3'C4UB4W":;0E*J%G QE)S,I2%&:<D>>Q!>WPGC*5M+&CEUBI2O#
$%! `.P``
`
end