Assume we are transmitting a real signal $x(t)$ through a channel $h$ in passband as :

$y(t) = x(t)cos(2πf_c t + ∅) * h(t)$ = $Re(x(t)e^{j2πf_c t})*h(t)$

where $f_c$ is the carrier frequency and $t$ is the time and $*$ is convolution operation. Normally, in idealistic conditions, that received signal $y(t)$ is real, but such phase offset is usually introduced due to the carrier frequency offset as described HERE, that phase offset can be represented in the received signal as :

$y_q(t) = x(t)cos(2πf_c t + ∅) * h(t) + α x(t)cos(2πf_c t +φ) * h(t) $

That means that we have phase shift φ affecting both real and imaginary part of the received signal; that makes the imaginary part of the received signal different than 0.

In that case, that phase offset can easily be estimated and compensated using $y_q(t)$, for example we can calculate $y_q(t) e^{-2πjφ}$ and then tune the parameter φ to have $sum(abs(imag(y_q(t)))^2) ≈0$, where $imag$ represents the imaginary part of $y_q(t)$. .Then we compensate that phase shift using that relationship. I think in that case the phase offset estimation will be very accurate as we estimated it based on every sample of the received signal.

My question, is the concept mentioned above correct ? Is there any similar algorithm/reference I can follow to understand that way well?

  • $\begingroup$ This is hard to follow since already your first equation seems to be wrong. Why is there a $j$ in the argument of the cosine ? In any case: all physical signals are real. Anything that's transmitted or received as a physical signal must be real as well. Complex signals are just a convenient way of modeling and/or processing real signals & channels. $\endgroup$
    – Hilmar
    Jul 6, 2021 at 8:07
  • $\begingroup$ @Hilmar Sorry for that error, I corrected it. Yes I totally agree with you that any transmitted or received signal must be real, but when converting from passband to baseband, such phase offset will be resulted leading to have complex signal even if the baseband transmitted signal is real too. here, I am transmitting a real baseband signal and I wanted to compensate that phase offset based on the yielded imaginary part. I hope it's clear now. $\endgroup$ Jul 6, 2021 at 8:29
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    $\begingroup$ Both real and complex signals are our way of modeling the physical world but one isn’t any more “real” than the other. “Real” and “Imaginary” are just an unfortunate naming but just because we need two scope probes to measure a complex signal does not make it any less “physical” than a real waveform. Similarly we would need two independent propagation paths between transmitter and receiver to transmit a complex signal which is unlikely but more likely is that we use a quadrature splitter in the receiver to create I and Q paths and what can be equally represented as a single complex waveform $\endgroup$ Jul 6, 2021 at 10:08
  • $\begingroup$ @DanBoschen But I mean when transmitting a passband signal in real communication, we receive a real signal too which means that effect of channel is real too, and the effects of imaginary part is coming from the CFO . is that right? (I mean if the baseband signal is real) $\endgroup$ Jul 6, 2021 at 12:14
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    $\begingroup$ The imaginary components you refer to are only in the complex baseband equivalent signal and can come from CFO or an assymetric channel as I provided with further details to your previous question (such as if the upper sideband of your spectrum is attenuated more than your lower sideband- there is no CFO in that case). The passband signal is represented by a real waveform and the baseband equivalent by a complex (which can be all real if we add no CFO or channel distortion)- I simply object when we say “real” is more physical than “complex”. Both are math representations of the physical $\endgroup$ Jul 6, 2021 at 12:19

1 Answer 1


To first clarify, this formula referenced by the OP is not showing the introduction of carrier frequency offset but rather is demonstrating multipath fading (the formula shows two different propagation paths each with a different phase offset and amplitude coefficient):

$$y_q(t) = x(t)cos(2πf_c t + ∅) * h(t) + α x(t)cos(2πf_c t +φ) * h(t) $$

Phase offset in the receiver is introduced due to these primary factors:

  • time delay between transmitter and receiver
  • Frequency offset when the transmitter and receiver run off of different clocks
  • Doppler frequency offset when there is any relative motion between transmitter and receiver

Ultimately all three of these effects contribute to phase offsets between the transmitted signal and the received signal, in addition to the multipath effects of the channel as demonstrated in simplest form with two paths by the initial formula referenced. The latter two effects listed are typically referred to as being frequency offsets, noting that a frequency offset is a linearly changing phase with time (although all three will inevitably be time variant).

The algorithms to correct for this are typically part of the carrier recovery loop or carrier frequency offset correction in the receiver. The best algorithm to use varies for different modulations but some examples of phase estimation and carrier recovery for single carrier modulations are further referenced in these posts:

High modulation index PSK - carrier recovery

Phase synchronization in BPSK

Software PLL tracking of carrier frequency in bandlimited transmission

  • $\begingroup$ Thank you so much for your clarification. I will read the provided references and check if I got what I need .. $\endgroup$ Jul 6, 2021 at 9:54

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