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When considering a direct conversion SDR receiving or emitting architecture, we can "represent" the received or transmitted signal as the sum of two components, the I and Q component.

Each component, represented by a complex number in the "IQ plane", or by a magnitude and a phase, is then mixed with a sin and cos originating from a local oscillator.

It's common to say that the argument of this complex number represent the phase of the signal, but what phase exactly ? Is it the phase related to the local oscillator signal ?

The origin of my question is the following : when considering a pure sinusoidal signal, for example, the phase of this signal changes all the time, so, in the IQ plane, I reckon it would be represented as a circle centered around the origin. Yet, when considering BPSK, its constellation diagram is just two dots, as if the signal was not really sinusoidal (two discrete phase).

So, is the phase of an IQ "sample" really a phase or is it a phase shift?

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The phase and magnitude of each complex IQ Sample is indeed the phase and magnitude of the modulated waveform at a higher carrier frequency, relative to that carrier for any linear modulation.

This is easy to see mathematically as follows starting with a complex IQ sample at baseband:

$$x[n] = I[n] + jQ[n] = A[n]e^{j\phi[n]}$$

It is helpful to note that the general expression $Ae^{j\phi}$ can be represented as a phasor on the complex plane with magnitude $A$ and angle $\phi$, and it is related to the real and imaginary components given Euler's formula (and simple geometry) as:

$$ Ae^{j\phi} = A\cos(\phi) + jA\sin(\phi) = I + j Q$$

To modulate $x[n]$ with a carrier (see this post for details), we simply multiply with $e^{j\omega_c n}$ and take the real part:

$$Re\{x[n]e^{j\omega_c n}\} = Re\{A[n]e^{j\phi[n]}e^{j\omega_c n}\}$$

$$= Re\{A[n]e^{j\omega_c n + \phi[n]}\} = A[n]\cos(\omega_c n + \phi[n])$$

Also very helpful when understanding what "phase" actually is and what it isn't (a point often misunderstood by many when first introduced to DSP since traditional instruction of "frequency" and DSP concepts are unfortunately often first introduced with sinusoids) is Euler's relationship for a sinusoid as:

$$\cos(\omega t) = \frac{1}{2}e^{j\omega t} + \frac{1}{2}e^{-j\omega t}$$

and from that we see how a REAL cosine (for which all samples have a phase of either 0 or 180 degrees only!) is composed of two COMPLEX phasors rotating in opposite directions such that angles cancel when summed.

To be clear on this, consider the time domain function $x(t)=\cos(\omega t + \phi)$. The function $x(t)$ is real valued, and with that every value for $x(t)$ has a phase of either 0° or 180° regardless of what $\phi$ is. As we learn in geometry, the cosine operator maps a rotation on the complex plane (which is phase) to its real component. Thus the phase angle $\phi$ will rotate each of the two phasor components (which themselves indeed have a phase as a complex quantity) according to:

$$\cos(\omega t+\phi) = \frac{1}{2}e^{j\omega t+\phi} + \frac{1}{2}e^{-j\omega t-\phi}$$

Phase is a rotation on the complex plane (whether it be a sample in the time domain as discussed here, or a sample in the frequency domain as we would get in the result of the FFT). Phase is NOT time delay, although the two are related. A delay in time is a linear phase in the frequency domain, meaning the phase of all samples in the Fourier Transform of a delayed time-domain signal will negatively increase proportional to the frequency. This leads many to associate phase directly with time delay given graphics introducing phase using two sine waves offset in time. This is very misleading and is the purpose of my DC battery with phase question to provoke this thinking; If you immediately think constructing a DC source in the lab with a phase is impossible, you may be confusing phase with time delay!

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