# Is sparse signal in time domain is sparse in Frequency domain

In OFDM system, if we have a multi-path environment, we supposed to have a sparse received signal in time domain, then $$FFT$$ is performed to convert it into frequency domain.

My question, does the results of FFT in frequency domain must be sparse since its equivalent time signal is sparse too ?

thank you.

• why does multipath propagation infer the sparseness of time-domain received signal? Because of fading (Rayleigh, Rician, Nakagami models, ...)? – AlexTP Sep 29 '18 at 7:32
• Yes because of Raleigh models, and because we have multi-path environment, we supposed to have sparsity in time domain – Fatima_Ali Sep 29 '18 at 7:51
• well I am not very sure about the fact that fading makes received signal sparse. But if your channel allows the Rayleigh-like discrete time channel models, with classic OFDM schema, I can say that the FFT signal can be modeled as signal passing a one tap channel which is also Rayleigh-like. And your definition "sparseness" is preserved. See my math dsp.stackexchange.com/questions/40157/… – AlexTP Sep 29 '18 at 8:02
• I would like to point out that a sparse signal in time would have a dense representation in frequency and vice versa (pulse train of course is different story) – MimSaad Sep 29 '18 at 14:58

My question, does the results of FFT in frequency domain must be sparse since its equivalent time signal is sparse too ?

No. In fact, it does the opposite: signals sparse in one domain are typically dense in the other.

If you want so, that's happening for the same reason that we have Heisenberg's Uncertainty Principle: Operators like the DFT don't allow us to preserve correlation / concentration of energy when going from one domain to the other (or back).

Let me comment on your original statement:

In OFDM system, if we have a multi-path environment, we supposed to have a sparse received signal in time domain, then FFT is performed to convert it into frequency domain.

That is false. The characterizing property of multipath environments is that they convolve the transmit signal with a nontrivial impulse response.

That converts even very sparse time-domain signals (e.g. a single dirac impulse) to dense ones. The only case where a channel would introduce sparsity would be if the the impulse response of the channel happened to be a matched filter to the transmit waveform.

But that can not be the case for an OFDM system, as that would directly contradict the principle of OFDM which is that you transport different bits of information on different frequency bins.

You might be considering a predistorting system, but then you'd mention that, and also, you'd know about the spectral properties of your signal (and hence could've negatively answered your original question yourself).

So, it feels like you've either taking something completely out of context, or you're misunderstanding something fundamentally.

• "No. In fact, it does the opposite: signals sparse in one domain are typically dense in the other". .. thanks for this answer. that what I need – Fatima_Ali Oct 1 '18 at 13:31
• I doubt it is. Your question feels way too much like you're taking something completely out of complex and expect complex answers to basic questions, and trivial answers to complex questions. – Marcus Müller Oct 1 '18 at 13:34
• Read the question again to be sure .. Then, I said it's ok, no need for wasting time and more useless chat – Fatima_Ali Oct 1 '18 at 14:28
• I read the question very carefully and that's why I have my doubts. – Marcus Müller Oct 1 '18 at 14:38