## Recover signal from the real part of IFFT

Started by 6 months ago●6 replies●latest reply 6 months ago●300 viewsblock, with the data on each carrier index placed on the corresponding input of the IFFT with all other

inputs set to zero. An IFFT shall be performed, and the real part of the output time sequence shall be taken, with any imaginary components discarded.

Hi,

IFFT of a spectrum with the 'upper' half of the frequencies zeroed out results in a complex time signal, called 'analytic' signal with 90° phase offset between real and imaginary part ( as to be exspected for a complex signal ). If you throw away the imaginary part and perform a FFT on the mere real valued signal, the 'lower' half of the spectrum is the same ( up to a constant factor ) as the spectrum you started from, the 'upper' half is no more zero but the conjugate complex of the 'lower' half. You loose no information.

Cheers

Detlef

##### Hi Frank_os,

##### The spectrum of any real signal has a transform which is Hermetian symmetric. This means real part is even symmetric and the imaginary part is odd symmetric. If you have the positive frequency terms you know the negative frequency terms: they are the conjugates of the positive frequency terms. Simply reflect the positive frequency bin (1:N/2 -) terms about the DC bin (don't include DC, bin 0, in the reflection and copy the conjugate terms into bins (-1:-1:-N/2+1). when you take the full length inverse transform you will get a real signal with computational noise in the imaginary array. Now you can discard the imaginary array.

##### fred h

fred and Detlef, thanks for replies. In the content of IEEE 1901.2, the spectral domain signal can be complex, such as DQPSK, 8DPSK or 16 QAM.

In IEEE 802.11a standard, after IFFT, the complex time domain signal is transmitted and it is easy to understand to use FFT to recover the frequency domain signal at the RX.

For IEEE 1901.2 system, the sub-carriers from 105-255 are zeros, the complex signal can be recovered.

Haven't heard this scheme before but my view is as below:

In effect the Tx generates symmetric spectrum since only one side of ifft is populated. Unlike other ofdms where both sides are populated.

Thus I/Q is there at Tx before ifft but the pair is used to populate one side only, at expense of wasting half spectrum in the channel.

The Rx is meant to recover the I/Q as usual knowing that it receives symmetric band and can ignore one half.

Notice that I/Q concept is separate from symmetry of spectrum.

Also, ultimately, the transmitted signal is a purely real signal, sampled in the time domain, thus all those complex parts are mathematical artefacts with various symmetries and are then exploited by the particular standard.

Aside: In the broad band time domain there is no "I/Q" pair, because there is no reference signal to do the separation. It's easy to fall into the I/Q trap, expecting to find that reference (i.e. in the `DC` band there is no Quadrature component, it's always "+i.0").

Yes, the Hermitian symmetry that Fred mentioned, in general, does not necessarily apply to OFDM where both the input samples to IFFT and the OFDM symbol (i.e., the IFFT output) can be complex.

The reason the imaginary part of IFFT can be discarded in IEEE 1901.2 is what Detleft said along with the fact the upper half subcarriers are zero'ed out.