# channel estimation based on time-domain pillots

I am confused if it's possible to estimate the channel based on the time-domain pilots. I mean I have such signal $$x$$ with length $$N$$ after taking its $$iFFT$$ we get $$X = ifft(x)$$. I have perfect knowledge on pilots signal in time domain, I mean I know what is $$X_{1:4:N}$$ (Again, it's in time domain after taking the ifft). The signal $$y = h*X$$ is the received signal representing the convolution between signal $$X$$ and channel $$h$$. Taking the $$FFt$$ to signal $$y$$ resulting $$Y = fft(y) = fft(X)*fft(h)$$.

My issue, if it's possible to estimate the channel in this case based on the pilots known in time-domain which are $$X_{1:4:N}$$ ?

• Please help me understand what you mean by you know only time-domain pilots $X$. Don't you also know frequency-domain $x$, if you know time-domain $X$, they are just FFT-IFFT pairs, right? And, if you know the frequency domain $x$, then you can use the typical channel estimation techniques for OFDM. I am assuming your system is OFDM based. Jun 22 '20 at 13:41
• Thank you for your reply, I mean I know the pilots which are $X_{1:4:N}$ after taking the $iFFT$. (let's say I inserted them after taking the iFFT). So I was asking if possible to estimate the channel based on pilots known for me in time-domain. To give you more details, I started multiplying my data $x$ with hartley matrix and then with iFFT matrix. so the resulted matrix is $X$, I want to do those two multiplication once, so I'm looking for a way to estimate the channel in that case. Jun 22 '20 at 14:32
• @Fatima_Ali if this is an OFDM system, you mustn't simply insert pilots in time domain! That breaks your OFDM signal! Jun 23 '20 at 9:03
• @MarcusMüller OK got it, .. if we ignored the OFDM system for the moment, can we estimate the channel $h$ based on $y$ in my above mentioned system? Jun 23 '20 at 12:57

if we ignored the OFDM system for the moment, can we estimate the channel h based on y in my above mentioned system?

Not really, because what you're describing is like if you could sample the channel, but at (symbol rate / 4), if I understood your approach correctly.

Hence, you get aliasing in your channel estimate, if it changes faster than (symbol rate / 4).

Your system, however, sees a channel that has full symbol rate (at least!) in bandwidth.

If your channel is slow enough, i.e. it's coherence time $$T_C > 4T_S$$, then this subsampling can be OK.

But you haven't given us any info that says it is (if you really ask us to ignore the OFDMness of it all).

If, on the other hand, we don't ignore the fact that this is supposed to be an OFDM system (in which case you really mustn't insert pilots in time domain, that breaks OFDM completely!), then your pilots are far too closely spaced, assuming your OFDM system has more than $$N=4$$ subcarriers: To use OFDM, your subchannels need to be coherent for one OFDM symbol duration $$T_{Sym}=NT_\text{sample}=\frac{N}{f_\text{sample}}$$.

I think you might be confusing something. OFDM systems of course use pilots on their subcarriers, which is in frequency domain, that they replace data symbols with every couple of OFDM symbols. But that means the pilots need to be inserted into your data stream, not into your time domain signal.

• I got it, thank you Jun 24 '20 at 14:02