How do we impose Hermitian Symmetry in optical OFDM?

Question

I’m currently trying to wrap my head around optical OFDM. I’m reading various tutorials and articles, but I’m hitting a stumbling block trying to understand Hermitian Symmetry and how it is applied. I’m taking the block diagram of an OFDM system from Figure 3 in this paper, and working through it step by step starting from the input to the transmitter. It’s probably best that I go step-by-step up until the point where I get confused.

Step 1 - serial to parallel

For the sake of simplicity I’m skipping the “Coding” and “Interleaving” steps from that paper. Let’s start with a serial bit stream we want to send: 1, 1, -1, -1, 1, 1, 1, -1, 1, -1, -1, -1, -1, 1, -1, -1, -1, 1…etc.

We will map this to parallel bit streams, one for each subcarrier. I’m using 4 subcarriers in this example, C1 to C4. The parallel streams are now:

C1	C2	C3	C4
1	1	-1	-1
1	1	1	-1
1	-1	-1	-1
-1	1	-1	-1
-1	1	1	-1
-1	-1	1	1

Step 2 - QAM modulation

Modulate each subcarrier. Here I'll use 4-QAM. Taking 2 bits at a time from each of the subcarrier columns I get:

C1	C2	C3	C4
1 +1j	1 +1j	-1 +1j	-1 -1j
1 -1j	-1 +1j	-1 -1j	-1-1j
-1 -1j	1 -1j	1 +1j	-1 +1j

Step 3 - IFFT

This is where my understanding starts to break down. I understand for “normal” OFDM we’d pass these values to the IFFT to generate the time-domain OFDM signal. However, what exactly are we sending to the IFFT block? Do we send it each row of that table in turn, i.e. the values of each subcarrier at a particular sample time?

Step 4 - Hermitian Symmetry

This is where my understanding really breaks down! I understand that Optical-OFDM require purely real-valued time-domain OFDM signals. I also follow that if the input to the IFFT block has the property of "Hermitian symmetry", then the IFFT output will be the desired real signal.

Hermitian symmetry is "imposed" on X_k symbols by ensuring that:

X₀ = X_N/2 = 0 and
X_k = X^*_N-k for 0 < k < N/2

Where N is the IFFT length and k is the carrier number (I assume?)

My confusion here is if we have to "impose" that certain carriers must take a given value (e.g. carrier 0 and carrier N/2 must equal 0) then isn't that somehow "overwriting" the value for that carrier that we have applied at the QAM modulation stage? That seems like it must be wrong to me, but I can't understand why!

So, apologies for the long-winded question, but just to clarify I'm asking if my understanding of the IFFT at Step 3 is correct, and what exactly we are doing when we "impose" Hermitian Symmetry onto our signal prior to the IFFT in Step 4.

Answer 1

the answer is actually very simple. Let us consider an IFFT size of N=32. In order to make sure that the output of the IFFT operation is real-valued (real numbers without imaginary part). y=IFFT(x) You have to follow the following arrangement to the input of IFFT operation (input is x): x(0)=0 x(N/2)=0 x(n)=conj(x(-n))

Example in Matlab:

M = 16; % For 16-QAM
IFFT_size = 32; % IFFT size
Nsc = IFFT_size/2-1; % number of data carrying subcarriers
data = randi([0 M-1],Nsc,1);
x = qammod(data,M);
x=[0;x(1:end);0;conj(x(end:-1:1))];
y=ifft(x);

Answer 2

I don't understand your question about the IFFT step: as in "normal" OFDM, you send a full vector to the IFFT, as that is a vector-valued operation of vectors. The output are (interpreted as) time-domain samples.

In fact, you really shouldn't have put Step 3 and 4 as separate operations, and certainly not in that order: the input to the IFFT has to be hermitian symmetric to yield a real-valued signal (which is what you need if you don't want to implement a quadrature upconverter to mix your signal into a carrier frequency, so in all honesty, fig. 3 is misleading in that the $-\sin(2\pi f_c t)$ branch doesn't have anything to do if you ensure your IFFT input is hermitian symmetric.)

My confusion here is if we have to "impose" that certain carriers must take a given value (e.g. carrier 0 and carrier N/2 must equal 0) then isn't that somehow "overwriting" the value for that carrier that we have applied at the QAM modulation stage?

No, you're just mixing up what you're doing: you're not free to assign an arbitrary value to each subcarrier at will, you need to make sure the vector is hermitian symmetric. You hence only have half the subcarriers to choose freely, the other one is inherently set to the complex conjugate.

So, you're not "overwriting" anything. You're just only assigning your QAM symbols to half the subcarriers.

Answer 3

Hermitian symmetry is "imposed" on X_k symbols by ensuring that:

$$x_0 = x_{N/2} = 0, \\x_k=x_{N-k}^* \forall k\in \left(k, \frac{N}{2}\right)$$

By this definition, for $N$ odd, you can choose $\frac{N-1}{2}$ independent values of $x_k$, and for $N$ even, you can choose $\frac{N}{2} - 1$ independent values of $x_k$. So you don't choose $N$ values and then wonder what to do with the extra $\frac{N}{2}+1$ values that don't fit -- you just start by choosing what will fit. To keep your brain from exploding, just choose the first such subcarriers, and set the higher-frequency subcarriers to their complex conjugates.

I would note, however, that to keep the output real, you don't have to set the first and last subcarrier to zero -- you only need to keep their imaginary parts zero. So at the cost of using two different constellations, you can choose

$$x_0, x_{N/2} \in \mathcal R, \\x_k=x_{N-k}^* \forall k\in \left(k, \frac{N}{2}\right)$$

Note that this is not necessarily a good idea -- it adds complexity to the subcarrier generation, and keeping $x_0 = 0$ lets you AC-couple the signal at the receiver, which in turn means you don't have to worry about DC bias in the receiver. But it's there if you want to add the accompanying complications.

(Actually, putting a pilot tone onto $x_{N/2}$ may be quite helpful).