Hi, A signal is composed of the sum of two sinusoids, and none of them has a period matching the transform size. There is DFT leakage after the FFT. Is there any other solution than windowing? Thank you.

# DFT Leakage

Started by ●October 11, 2012

Reply by ●October 11, 20122012-10-11

On Thursday, October 11, 2012 9:21:57 PM UTC-4, commsignal wrote:> Hi, > > A signal is composed of the sum of two sinusoids, and none of them has a > > period matching the transform size. There is DFT leakage after the FFT. Is > > there any other solution than windowing? > > Thank you.I'm not real sure of your question. Are you asking how to find the amplitude coefs for just two frequencies and hoping windowing will make your estimate better? If there is any or much noise in your two signals? If not, a simple 2 x 2 matrix problem on the coefs will yield your values. If you have a lot of noise, the issues are much harder. Please clarify your problem and we can likely help you more. Clay

Reply by ●October 11, 20122012-10-11

>On Thursday, October 11, 2012 9:21:57 PM UTC-4, commsignal wrote: >> Hi, >>=20 >> A signal is composed of the sum of two sinusoids, and none of them has=>a >>=20 >> period matching the transform size. There is DFT leakage after the FFT.I=>s >>=20 >> there any other solution than windowing? >>=20 >> Thank you. > >I'm not real sure of your question. Are you asking how to find theamplitud=>e coefs for just two frequencies and hoping windowing will make yourestima=>te better? If there is any or much noise in your two signals? If not, asim=>ple 2 x 2 matrix problem on the coefs will yield your values. If you havea=> lot of noise, the issues are much harder. Please clarify your problem and=>we can likely help you more. > >Clay >Actually I was thinking of taking e^(j omega_1 n)*(a QAM signal) + e^(j omega_2)*(another QAM signal) into frequency domain and trying to see what happens there. But omega_1 and omega_2 destroy the periodicity. Can their effects be removed, even in case I know omega_1 and omega_2?

Reply by ●October 12, 20122012-10-12

On 10/11/12 9:48 PM, commsignal wrote: ...> Actually I was thinking of taking > e^(j omega_1 n)*(a QAM signal) + e^(j omega_2)*(another QAM signal) > into frequency domain and trying to see what > happens there. But omega_1 and omega_2 destroy the periodicity. Can their > effects be removed, even in case I know omega_1 and omega_2?so you want to look at the spectrums of the two QAM signals and are worrying how much the window leakage (remember, no window means the rectangular window) of one interferes with the other? if the windows are well defined, you can actually predict how much spectral leakage of one modulated QAM is at any other specific frequency. it might not be impossible to detangle two. it's like two equations and two unknowns (the values of the two QAM signals). but usually we want to minimize the spillage from one to the other, so that's when windowing is often applied. the window most people use to control spectral leakage is the Kaiser since it gives you a pretty dependable tradeoff control trading off the sharpness of the window's main lobe with the spillage amount. i think the parameter is called "beta". it's in O&S. how far are omega_1 and omega_2 apart in DFT bins? the size of the window in bins (or samples) and the DFT length are the parameters that determines how wide (in samples) the window needs to be. -- r b-j rbj@audioimagination.com "Imagination is more important than knowledge."

Reply by ●October 12, 20122012-10-12

Hello, more often than not, your channels are arranged on a grid that allows integer multiples over a realistic measurement length. Let's say, I want to observe 1 ms of signal. A typical channel raster is 200 kHz. In other words, any frequency offset that is allowed by some hypothetical radio standard comes in multiples of 200 cycles over my observation length => no problem. With a bit of "frequency planning", you can set up a simulation with cyclic signals and avoid FFT windowing completely. If for some reason the numbers don't work out, it's usually no problem to tweak them a little until they do. For example, dealing with OFDM symbols, I adjust my cyclic prefix lengths so that I get -exactly- 14 symbols in 1 ms.

Reply by ●October 12, 20122012-10-12

On Thu, 11 Oct 2012 20:21:57 -0500, "commsignal" <58672@dsprelated> wrote:> A signal is composed of the sum of two sinusoids, and none of them has a >period matching the transform size. There is DFT leakage after the FFT. Is >there any other solution than windowing?"Notch Fourier Transform"; Yoshiaki Tadokoro and Kenichi Abe, IEEE TRANSACTIONS ON ACOUSTICS, SPEECH, AND SIGNAL PROCESSING, VOL. ASSP-35, NO. 9, September 1987.