# Revert anayltic signal(length(N/2)) to real signal (length(N))

As a beginner in signal processing, i'll try to explain my problem most thoroughly. I'll firstly explain the idea of the transformation the real valued data has undergone. This includes the ideas and reasons why it does work and the way to revert the procedure. Then i'll explain my difficulties in reverting it.

First of all a property of the FFT of a real signal:

Given an $N$ data point real valued signal $x(t)$, it has $\frac{N}{2} + 1$ unique Fourrier coefficients $X(f_i)$ with $i \in {0,..,N-1}$. The other $\frac{N}{2}-1$ coefficients can be retrieved by inverting the spectrum of the coefficients $X(f_1)$ to $X(f_{\frac{N}{2}})$. So in some sense all the information of my signal is stored in the first $\frac{N}{2}+1$ of it Fourrier coefficients. Keep that in mind, because this property i'm going to exploit to retrieve the original signal from it's analytic representation.

Now to the theory of the procedure the data has undergone:

Let $X_a (f_i)$ be the analytic representation of the Fourrier coefficients $X(f_i)$ of $x(t)$. Then $X_a(f_i)=0$, if $f_i < 0$, and for the Fourrier Coefficients of the original signal $x(t)$ we know that $X(f_i)= \frac{X_a(F_i)}{2}$with $i \in 1,...\frac{N}{2}$. So $\frac{N}{2}-1$ coefficients carry no additional information about the signal. This allows a downsampling to $\frac{N}{2}+1$ frequencies, by cutting those $X_a(f_i)$ off, with $f_i <0$. And no information is lost, as the original signal was real. To make sure you can follow me http://en.wikipedia.org/wiki/Analytic_signal explains it in detail. Let's call the resulting complex signal of length $\frac{N}{2} +1$ $x_c(t)$.

The way to retrieve the original real signal would be:

If i'd want to retrieve my original signal all i'd have to do is invert the Fourrier Cofficient vector $[X_a(f_i)]$, $i \in 1,N-1$ and append them to my frequency spectrum. Resulting in the spectrum: $[X_a(f_0),...X_a(f_{\frac{N}{2}+1}), X_a(f_{\frac{N}{2}}),X_a(f_{\frac{N}{2}-1}), X_a(f_{\frac{N}{2}-2}),...X_a(f_1)]=FFT(x(t))$ of my original signal $x(t)$.

Now to my situation:

I do get something similar to the results of above modification of signals. The major difference is that i don't get all $\frac{N}{2}+1$ coefficients of the complex signal $x_c(t)$. But what i do get is the complex signal from $0$ to $\frac{N}{2}-1$. So what i'm lacking is the coefficient $X_a(f_{\frac{N}{2}})= \sum_{k=0}^{N} e^{i\pi k}x_a(k)$ of the Fourrier transform of my Fourrier pair $(X_a(f_i),x_a(t))$. This is a real valued Fouerrier coefficient of the signal $x(t)$ at the nyquist frequency.

PS: Hope this format is way better than before.

• Hello Marcel. I am very, very confused by your question, so it is hard to answer. Can you cull it down and make it succinct please? Thanks. Commented Aug 8, 2013 at 14:01
• I'll try to make it short and more meaningful user4619. Commented Aug 8, 2013 at 15:13
• I did it and do hope it is more useful now.. Commented Aug 9, 2013 at 8:22