# OFDM Signal and IQ Paths

I understand that we have analog complex signal after it passes DAC after coming out of IFFT in OFDM Transmitter. After that we do orthogonal upconversion on it and send it. I've question now for OFDM Reciever, suppose i recieved a signal then i separated I & Q component, for processing anything on the signal now I've to do processing separately on I & Q Paths (independent paths) does the orginal OFDM Signal even has any significance until i combine I & Q back at input of ADC ? because before ADC everything is split at recieved end in I & Q path and processing is done on it. Little confused, may be I'm thinking too much, pls guide.

• I guess you know how to down convert the received RF signal to baseband and how to separate I and Q. Then you combine them into a complex signal and do ADC (or you do ADC on each of I and Q and then combine them into a complex signal). After that, do FFT on the combined complex signal to recover original modulating symbols used in the OFDM modulator in the transmitter. Jun 19 at 14:18
• @user295357 Suppose at the transmitter part, i have following equation x(t) is complex baseband signal and w is rf frequency x(t) with e^jwt = i(t) cos wt - q(t) sin wt + j q(t) cos wt + j i(t) sin wt, and then i pick real part of it and send i(t) cos wt - q(t) sin wt = r(t). Now at reception end I'll again do the following r(t) * e^ jwt it will yield real part having cos square wt and imaginary part sine square wt, how do we filter real and imaginary part? Can you show the maths pls Jun 19 at 18:03
• See my response as an Answer since it is too long to be a comment. Jun 19 at 20:17

@Reddi Suresh
Denote the received RF signal as $$r(t)$$. In the receiver, do following three things for down-converting and recovering $$i(t)$$ and $$q(t)$$.

(1) Multiply $$r(t)$$ with $$2cos(\omega_0t)$$, that is,

$$c(t) = 2r(t)cos(\omega_0 t)=2i(t)cos^2(\omega_0t)-2q(t)sin(\omega_0t)cos(\omega_0t)$$

$$c(t)=i(t)[1+cos(2\omega_0t)]-q(t)sin(2\omega_0t)$$.

(2) Multiply $$r(t)$$ with $$[-2sin(\omega_0t)]$$, that is,

$$s(t)=-2r(t)sin(\omega_0t)=2q(t)sin^2(\omega_0t)-2i(t)cos(\omega_0t)sin(\omega_0t)$$

$$s(t)=q(t)[1-cos(2\omega_0t)]-i(t)sin(2\omega_0t)$$.

(3) Passing each of $$c(t)$$ and $$s(t)$$ through a low-pass filter to remove all components whose frequency higher than $$\omega_0$$, you get $$i(t)$$ and $$q(t)$$ respectively.

Equivalently, if you like, you can expresse the recovered $$i(t)$$ and $$q(t)$$ as a complex signal as

$$i(t)+jq(t)=LPF\{2r(t)e^{-j\omega_0t}\}$$

where $$LPF\{.\}$$ stands for low-pass filtering. This can be easily verified based on above equations.

This answer assumes a clear understanding of "negative frequencies" and how it relates to real and complex signals. If that concept is not very clear, reviewing this post first may be helpful. Please also see this post that is directly related to the OP's question.

I think the confusion is with "Separating into I and Q components". There really isn't a "separation" but using complex representation to represent the signal. For real signals at baseband, the spectrum centered about DC is complex conjugate symmetric: the negative frequencies which represent the lower half of the spectrum is identical to the positive frequencies (which represent the upper half of the spectrum) other than phase reversal. They are completely redundant. Whatever is centered about DC for the baseband signal becomes what is centered about any carrier frequency that we translate the signal to in the up-conversion process. We care a lot about spectral occupancy, so that redundant spectrum if we used a real baseband signal means we use up twice as much spectrum as we need to. This is why most modern waveforms have a complex baseband spectrum: with that the positive and negative frequencies at baseband are completely independent (and therefore the lower half and upper half of the spectrum at any real carrier frequency it is translated to will be as well). This immediately doubles are spectral efficiency, which is a valued trade for the increase in hardware complexity. To do this just means the baseband signal (when centered at DC) must be complex. That one signal is represented by its real ("I" for in-phase) and imaginary ("Q" for quadrature) components-- it is still one signal, we just need two data paths to represent it. Just as we need two real numbers on paper to represent a complex number using either cartesian or polar coordinates.

A main point is summarized with the graphics below comparing spectrums for real and complex baseband waveforms (centered at DC). In order to have a spectrum that has independent positive and negative frequencies (as we see on the right), the waveform must be complex. If we frequency translate both of the waveforms given above to a real carrier frequency, we get the resulting passband spectrums shown below (showing the magnitude only for simplification but the phase would also translate accordingly): The difference in implementation for translating a baseband signal (whether it be real or complex) is shown in the simplified diagrams below (filtering to make this complete is not depicted). This shows translating from baseband to an RF or IF carrier frequency. The reverse process for received (translating from RF or IF to baseband) would be similar in reverse form: So in short we take an IFFT of the data in the transmitter to get a complex baseband signal. We frequency translate that to a real carrier. In the receiver we do the opposite: we frequency translate the waveform from a real carrier to complex baseband, then with the complex baseband signal we take the FFT to recover the data (simplified as there are a lot of details with framing, carrier and timing recovery etc in the receiver).

Where we do the ADC and DAC in the above process can be anywhere technology allows (we can do baseband sampling with two ADC's or RF/ IF sampling of the real signal at a carrier with one ADC for example), it doesn't change the processing, it just changes whether the signal is represented in analog or digital form.

For further details on quadrature (IQ) frequency translation, this post may be of interest.

• Anyway, $i(t)$ and $q(t)$ carries independent information, thus at some stages of processing, they have to be separated, otherwise the transmitted information would not be useful to the destined user. As long as $i(t)$ and $q(t)$ finally can be correctly recovered (including being correctly separated) finally in the receiver, there is no essential difference whether they are viewed as a single complex signal or two independent real signals at each stage. From another point of view, the benefit to express them as a complex signal is to make them separable. Jun 21 at 20:48
• It is true that with an RF frequency band, if only the cosine carrier is used to transmit a real baseband signal, the band is not efficiently used, since the sine carrier is not used to transmit another independent real baseband signal with the same RF band. Both cosine and sine carriers can be used because they are orthogonal, so that the two real baseband signals can be separated. Another possible way to avoid waste of the bandwidth when only one RF carrier is used is to transmit single-side-band (SSD) signal based on the Hilbert transform theory. Jun 21 at 22:01
• @user295357 Respectfully I don't see your point (yet). I do note that there is no way to express a complex number without using two real numbers so we always "separate" on paper and implementation- but still it is always one complex waveform and it greatly simplifies are thinking to see it that way (in my opinion). Jun 22 at 2:09
• Also single-sideband and Hilbert transform theory is all well described with complex numbers, I don't really see it as a different approach. For me using complex number representations greatly simplifies the processing. (We can of course refuse that approach and describe all signal flows as separate paths with sines and cosines; I've done both approaches and am sold on the utility of complex representations- bottom line is either are just that: the representations we use. This may help: mriquestions.com/uploads/3/4/5/7/34572113/… Jun 22 at 2:12
• Yes and I totally get yours....there's no "right answer" just different ways of looking at things and what "reality" means to us. I had figured you meant SSB and it was a nice example. Jun 22 at 13:58

Note that the IFFT sequence for one set of complex numbers (vectors) transmitted through a communications medium is not periodic. So that single transmitted complex time-domain sequence is not going to be 'OFDM' because it is not a periodic sequence. This means that this taught IFFT method for OFDM generation should be taken with a grain of salt. The transmitted sequence is actually just I/Q transmission (quadrature modulation). You would only get 'OFDM' if each IFFT sequence is periodic, and that is not going to be possible due to needing to send lots of different IFFT sequences in succession. So the IFFT transmission does not produce a physical OFDM spectrum at all. That is, the transmitted IFFT 'signal' is not even an 'OFDM' signal as such.