Is it correct to say that the spectrum of an OFDM signal is rectangular pulse?
What about the spectrum of each transmitted signal over subcarriers?
If the OFDM signal has a rectangular frequency domain spectrum, then the time domain after IDFT is much like a sinc function. Do we need pulse shaping in this case, since sinc function already looks like pulse shape?
Your statement isn't strictly correct; the spectrum of an OFDM signal isn't a rectangular pulse. Think about it this way: OFDM is really just a method to efficiently modulate multiple low-rate narrowband signals that are sent simultaneously. Therefore, the OFDM signal's spectrum is really just the sum of the spectra of each of its narrowband components.
For example, if your OFDM signal has 64 BPSK subcarriers, then its spectrum is equal to the spectra of 64 narrowband BPSK signals, spaced at the OFDM subcarrier spacing. If you were to estimate the OFDM signal's power spectrum with a wide enough resolution bandwidth, then the spectrum would appear to be rectangular. However, if you look closely enough at the signal's spectrum, you'll notice that it isn't flat.
Here is an example MATLAB script that demonstrates the phenomenon:
% generate random BPSK symbol vector; this is equal to the "low-resolution"
% spectrum, calculated with bin spacing synchronized with the OFDM
% subcarrier grid
X = 1 - 2 * (randn(64,1) > 0.5);
f = -0.5 + (0:length(X)-1) / 64;
% create time-domain OFDM symbol using the IFFT
x = ifft(X);
% calculate interpolated spectrum to see what it looks like between OFDM
% subcarrier centers
Sh = fftshift(fft(x,4096));
fh = -0.5 + (0:length(Sh)-1) / 4096;
figure;
plot(f,abs(X)); hold all; grid on;
plot(fh,abs(Sh));
legend('Regular', 'Interpolated');
As the modulated OFDM signal doesn't have a perfectly rectangular spectrum as you guessed, it does not necessarily have a sinc-like shape in the time domain either.
