FFT Narrowband Reconstruction

Is it possible to reconstruct a narrowband, time domain signal from a
subset of wideband FFT output coefficients ?

Maybe a simple example will help explain:
Let's suppose I sampled a wideband set of signals (at different carrier
frequencies) and applied a 128K-pt FFT.  I know that I can get the
original wideband signal back out by taking the inverse FFT, howevever,
is it possible to reconstruct JUST ONE of the narrowband signals by
taking an IFFT of only a few of the output coefficients (eg, a signal
that spans just 16 coefficients).

>Is it possible to reconstruct a narrowband, time domain signal from a
>subset of wideband FFT output coefficients ?
>
>Maybe a simple example will help explain:
>Let's suppose I sampled a wideband set of signals (at different carrier
>frequencies) and applied a 128K-pt FFT.  I know that I can get the
>original wideband signal back out by taking the inverse FFT, howevever,
>is it possible to reconstruct JUST ONE of the narrowband signals by
>taking an IFFT of only a few of the output coefficients (eg, a signal
>that spans just 16 coefficients).
>

By just keeping the spectral complex pairs that comprise your narrowband
signals, and setting all other pairs to zero, then performing the 128k
IFFT, this should give you what you want.

Jeff

Jeff Caunter wrote:
> >Is it possible to reconstruct a narrowband, time domain signal from a
> >subset of wideband FFT output coefficients ?
> >
> >Maybe a simple example will help explain:
> >Let's suppose I sampled a wideband set of signals (at different carrier
> >frequencies) and applied a 128K-pt FFT.  I know that I can get the
> >original wideband signal back out by taking the inverse FFT, howevever,
> >is it possible to reconstruct JUST ONE of the narrowband signals by
> >taking an IFFT of only a few of the output coefficients (eg, a signal
> >that spans just 16 coefficients).
> >
>
> By just keeping the spectral complex pairs that comprise your narrowband
> signals, and setting all other pairs to zero, then performing the 128k
> IFFT, this should give you what you want.
>
> Jeff


Alas, I was hoping to find a way such that I didn't have to do the full
128k-pt IFFT.  Perhaps only the 16-pt IFFT.  Maybe with some
pre-twiddling or something ???

On 4 Nov 2006 16:44:43 -0800, "yazz35@gmail.com" <yazz35@gmail.com> wrote:

>Alas, I was hoping to find a way such that I didn't have to do the full
>128k-pt IFFT.  Perhaps only the 16-pt IFFT.  Maybe with some
>pre-twiddling or something ???

Well, it depends on what you mean by "reconstruct the narrowband signal".  If
you know in advance that the signal is a single sine wave, then you can
reconstruct it by finding the FFT bin with the highest power, interpolating the
frequency, and then synthesizing a sine wave.  Unless you need to know the phase
too.  I'm not sure how to combine bin interpolation and phase.  But if you don't
interpolate, then the phase is directly readable from the sine and cosine parts
of the bin with the most power.



Robert Scott
Ypsilanti, Michigan

yazz35@gmail.com schrieb:

> Is it possible to reconstruct a narrowband, time domain signal from a
> subset of wideband FFT output coefficients ?

Sure. Here is the DFT/IDFT defintion:

                                  N-1
    DFT{ x[n] }  =  X[k]  =       SUM{ x[n] * exp(-j*2*pi*n*k/N) }
                                  n=0


                                  N-1
    iDFT{ x[n] } =  x[n]  = 1/N * SUM{ X[k] * exp(+j*2*pi*n*k/N) }
                                  k=0

To resynthesize the x[n] via IDFT from just a few (or a single) of the
X[k], only compute the partial sum with the relevant coefficients.

Regards,
Andor