Setting the duration of subcarriers in OFDM system

Question

If have an ofdm system with $N$ subcarriers, and sampling frequency $F_s$, that means that duration of each subcarrier is $T_s = \ {1/F_s}$ and the duration of whole symbol (which includes $N$ subcarriers) will be $T_{sym} = N/F_s$. I think till now, it's ok.

Additionally, the $F_s$ is coming from $F_s = B*a$, where $B$ is the width of baseband signal and $a$ is the upsampling factor.

My concern is, what's about if I need to transmit $2N$ instead of $N$ subcarriers within the same symbol duration, which parameters I should change? Will that be correct to change the $F_s$. however if I set $F_s = 2F_s$, in that case it's like I doubles the width of baseband signal too, which is $B$!

Answer 1

I'll try it this way (the other answers are right, you really have a problem with basics):

OFDM works like this:

0. You have $N$ values. You want to put this $N$ values on $N$ subcarriers and transmit them at the same time.
1. You take the $N$ values, and apply an IDFT to it. An IDFT always gives you as much output as you give it input. So, you get $N$ values out. This is one OFDM symbol. It has a length of $N$, as you can see. Therefore, each value on each subcarrier has a "length" of $N$, too.
2. you transmit these $N$ values as samples¹. That means the time duration of each symbols is $N/f_s$, with $f_s$ being the sampling rate.

As you can see, length == number of subcarriers. There's no way around that. If you want $2N $ subcarriers, you will have a length of $2N$. That's how it is.

If your requirement is, as stated in your question:

My concern is, what's about if I need to transmit $2N$ instead of $N$ subcarriers within the same symbol duration, which parameters I should change?

So, you know that so, if the duration needs to stay the same, then

\begin{align} \frac{N_{old}}{f_{s,old}} &= \frac{N_{new}}{f_{s,new}}= \frac{2N_{old}}{f_{s,new}} \end{align}

So, let's figure out the value of the only thing we can change in this equation:

\begin{align} \frac{N_{old}}{f_{s,old}} &= \frac{2N_{old}}{f_{s,new}} &&\| :N_{old}\\ \frac{1}{f_{s,old}} &= \frac{2}{f_{s,new}} &&\| \cdot f_{s,new}\\ \frac{f_{s,new}}{f_{s,old}} &= 2 &&\| \cdot f_{s,old}\\ {f_{s,new}}&=2{f_{s,old}}.&&&\rule{0.5em}{0.5em} \end{align}

From that equation you can directly see that you need $f_{s,new}=2f_{s,old}$. There is no other way to keep the duration equal, but get twice as many subcarriers.

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¹ there's a guard interval, typically occupied by a cyclic prefix, that you'll have to add, usually. But we're not at a point where we need to discuss that.

Answer 2

You can decrease the subcarriers spacing which gives you longer ofdm symbol.

Answer 3

With 802.11ax versus 802.11ac specification: for 20MHz bandwidth, we went from 64 to 256 FFT (X4) -- however, the frequency separation went from 312.5KHz down to 78.125 (/4) and the symbol duration (w/o CP) went from 3.2 to 12.8 usecs (x4). These constraints are ALL related to the use of the FFT for the sub-carrier separation at the receiver. Maybe we need to rethink the use of the FFT?

Answer 4

You can do that by decreasing the sub-carriers spacing into half which is equivalent to increasing the number of sub-carriers transmitted in the same existed bandwidth.
For example, the bandwidth will be occupied by 2N subcarriers instead of N.

you can do that by only using 2N-IDFT OFDM instead of N-DFT OFDM, but to simulate that in MATLAB, you need to use a filter to included the effect of sub-carriers spacing such that: $F_{spacing}$ = $F_s/N$, so using $2N$ will automatically decrease the sub-carriers spacing.

Just use the same sampling frequency with different number of subcarriers to investigate the performance of that effect.

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Let $X$ be your OFDM signal vector after adding the guard interval. It means that you already performed the IFFT and added the CP guard interval. B is the bandwidth you want to transmit the data through it, it's a value, for example $B = 20$ x $10^6$
The signal $X$ has $N = 64$ sub-carriers and $N_g = 16 = N/4$ guard interval. (These are the paramters of WLAN standard, I think).

So, the IFFT/ FFT period is $3.2$ $us$ which is $T_{fft } = 1/(B/N)$ and the guard interval period is $T_g = T_{fft}/4$. Hence, the duration of OFDM symbol will be $T_t = T_g + T_{fft }$. I think till here, it's clear. The duration of your ofdm symbol is $T_s = T_t/(N+N_g)$.

let's have your carrier frequency $F_c = 2.412$ x $10 ^ 9$, you set the time vector by matlab e.g., t=T_s /50:T_s /50:T_s ; Then you perform your passband signal by matlab as follows:

Carr = [];
for n=1:length(X)
    Carr_real = real(X(n))*cos(2*pi*Fc*t); 
    Carr_imag = imag(X(n))*sin(2*pi*Fc*t);
    Carr = [Carr Carr_real+Carr_imag];
end 

Or, you can use the $F_s$ to set the vector t, such as t = 0 : 1/F_s:(1/F_s)*length(S)-1 $S$ is the signal $X$ after upsampling it. (It must be upsampled following your bandwidth and upsampling factor). Finally your up-converted signal Carr will be $Carr = S.$*$exp(2*pi*F_c*t)$, $.*$ is elements wise multiplication. Then you transmit the $S_p$ via the channel. This signal Carr is called the pass-band signal.

In that way, you will transmit different signals with different number of sub-carriers through the same bandwidth. You see? everything is related to each other and all parameters must be set following the system structures. Practically, all signals must be up-converted into passband before transmission, but baseband is usally used for simulation only. I don' think that you can do this procedure using other methods !!

if you got difficulties to implement that in MATLAB, let me know.

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Answer 5

Per definition of how OFDM works, you take a vector of input symbols, and interpret them as parallel sub carrier data. You then do an N-IDFT. That directly says the length of a symbol on a subcarrier is N samples. Always.

Of course, there might be cyclic prefixing, which increases the length of the OFDM symbol further.

what's about if I need to transmit 2N instead of N subcarriers within the same symbol duration,

Since 2N is twice as many as N samples, if the duration needs to remain the same, you will have to double your sampling rate - and thus your bandwidth. As the saying goes, there is no free lunch.

That's all very good and consistent: OFDM isn't magic - it can't produce spectrum out of nowhere. It's just a clever way of dividing the spectrum to make it easier to use. Nothing more, nothing less. You can't transport more symbols per second in the same bandwidth with OFDM than with a single carrier system; all you achieve is easier/better equalization.