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I'm learning about OFDM Transmitter and see that output of IFFT Block produces complex digital signal and we need DAC to convert it to analog and then use RF Frequency mixing to send it out of transmitter. But this RF Freq mixing is not simply multiplying signal with high frequency, rather I see something called as inphase and Quadrature phase, i don't understand this , can someone pls guide ? Why it is done and what's the use of it? Is it done to send a real signal out of transmitter?

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In OFDM, the output of DAC after IFFT is a complex analog signal. In order to transmit this complex signal, two orthogonal RF channels are needed, one to transmit the real part, another one to transmit the imaginary part. So, in practice, the cosine wave and the sine wave of a same RF frequency are used to respectively transmit the real part and imaginary part. Such kind of frequency-converter (also called frequency mixer) converting a baseband complex signal to a two-channel RF signal is called quadrature frequency converter (or mixer).

Expressed in math, if the baseband complex signal is expressed as

$$x(t)=i(t)+jq(t)$$

where $i(t)$ and $q(t)$ are often respectively referred to as the in-phase part and the quadrature-phase part of $x(t)$. The RF signal then is

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

Clearly, $r(t)$ is a real RF signal to be transmitted. Equivalently, $r(t)$ is often expressed as

$$r(t)=\Re e \big\{x(t)e^{j\omega_0t} \big\}$$

for convenience, meaning that $r(t)$ is the real part of the product of $x(t)$ and $e^{j\omega_0t}$. Indeed, this can be easily verified with simple math.

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  • $\begingroup$ I know there are some communications texts that capitalize the in-phase and quadrature baseband signals, but what if you want to depict the Fourier Transform of those signals? It just seems to me that it's better and more consistent with other signal processing literature to leave functions of time in lower case and their Fourier or Laplace Transforms of those show in upper case. Just my dumb opinion. $\endgroup$ Jun 16, 2023 at 0:32
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    $\begingroup$ Hi robert bristow-johnson Oh, I got your point. Sorry for I didn't think much about whether the capital letter or lower case letter should be used here to denote the in-phase and quadrature components of $x(t)$. When I was writing, the only thing made me a little bit hesitate was trying to avoid using the letter $i$ and $j$ in the same equation. It never means anything wrong to use $i(t)$ and $q(t)$ instead. Now I agree that $i(t)$ and $q(t)$ may be better. Thank you very much. $\endgroup$
    – user295357
    Jun 16, 2023 at 0:54
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    $\begingroup$ @robert bristow-johnson I accept your point, and made changes. Please see my edition with equations updated. Thanks. $\endgroup$
    – user295357
    Jun 16, 2023 at 1:04
  • $\begingroup$ Is it not possible to transmit complex signal? @user295357 $\endgroup$ Jun 17, 2023 at 11:10
  • $\begingroup$ If you have two independent channels for transmitting, then you can essentially transmit “a complex signal” to the same extent we can send a complex signal across a circuit board by using two traces labeled “I” and “Q” for in-phase and quadrature. I do find it very convenient to refer to those two real signals as one complex signal I+jQ. Just because we can do something doesn’t mean we should, so my comment is illustrating how it can be done only. Why we do or don’t could be worthy of a separate question. $\endgroup$ Jun 17, 2023 at 14:12
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"Mixing" has a different meaning for RF than it does for audio.

In audio mixing is a linear combination of the two (or more) signals. In RF, it's the modulation that bumps a baseband signal up to the RF frequency.

The reason they call it "mixing" in RF is that the way they created the cross-product term, $x(t) \cos(\omega_0 t)$ which is what we want for the RF modulated waveform, they first add the two signals and then pass that sum through a non-linear element that generates the cross-product term.

Now quadrature modulation is a slightly different multiplication. It like first , $$ x(t) = i(t) \, + \, j\, q(t) $$

where $i(t)$ and $q(t)$ are the in-phase and quadrature components and the RF output would eventually be the real part of

$$ x(t) \cdot e^{j \omega_0 t} $$

Now, if your DSP is fast enough, you can do that complex math in a DSP and output the real part as the composite RF signal. I normally think that most modulation to RF is done in the analog domain because the RF frequencies are often so high. 100s of MHz. But a DSP can easily modulate the baseband quadrature signal, $x(t)$, to an Intermediate Frequency (IF) which is much higher than baseband, but much lower than the physical RF frequency it will be bumped to when it's going out a microwave dish.

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  • $\begingroup$ Well, a very long time ago RF mixing was accomplished by adding the RF and LO and then running it through a nonlinear element. They started switching over to multiplying the RF and LO, to various degrees of perfection, back in the vacuum-tube days. $\endgroup$
    – TimWescott
    Jun 17, 2023 at 16:33
  • $\begingroup$ @TimWescott it was that way with my ham radio gear (ca. 1970 with vacuum tubes). First the audio was bumped to IF by adding the audio to the LO and then running that through a funky-biased tube circuit and then filtering the output so that only the cross-product term survived. That's how they did multiplication. They didn't have a Gilbert cell mulitplier or any log circuits. They had a non-linear element that had an $x^2$ term in it that generates the cross-product term. Then, after that, the IF is bumped up to RF in a similar manner. $\endgroup$ Jun 17, 2023 at 17:07
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    $\begingroup$ I didn't say it was common. Page 302, figure 12.10 of the 1956 ARRL Handbook shows a balanced modulator for SSB using both sections of a 12AU7. I didn't find it in a brief search of my library, but ca. late 40's through the 50's there was a twin-plate beam-deflection balanced modulator tube (so, the vacuum tube equivalent of half a Gilbert Cell). Somewhere in my stash of old ARRL Handbooks I have a build article on a SSB receiver using the phasing method (baseband I/Q for you young folks) that uses pairs of those tubes for double-balanced demodulation. $\endgroup$
    – TimWescott
    Jun 17, 2023 at 17:53
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    $\begingroup$ 6AR8 was used in TVs as a balanced modulator. The 7360 was similar, but optimized for SSB. Not mentioned in the cited articles is the fact (conveyed to me verbally by a friend ages ago) that these fell out of favor because they were susceptible to imbalance caused by magnetic fields. It's the 7360 that I recall seeing in a radio schematic. $\endgroup$
    – TimWescott
    Jun 17, 2023 at 18:01

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