Reading about complex signals, I learn a phasor is composed of real and imaginary components as $e^{2\pi j \cdot \omega}$ and $e^{2\pi ⁻j \cdot \omega}$. Why is this? How can a signal be imaginary?
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$\begingroup$ aitia, Perhaps the material at the following web page may help you: dsprelated.com/showarticle/192.php $\endgroup$– Richard LyonsCommented Nov 18, 2020 at 12:11
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$\begingroup$ It seems the formula is different, I will edit my question to reflect this. $\endgroup$– aitíaCommented Nov 18, 2020 at 15:01
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$\begingroup$ How can a number be imaginary ? $z = a + j b$ is a complex number with imaginary part $jb$... The imaginary unit is $j = \sqrt{-1}$ ? How can this all be? $\endgroup$– Fat32Commented Nov 19, 2020 at 10:43
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1$\begingroup$ Fat32. The adjectives "complex" and "imaginary" are certainly unfortunate. Rather than "complex", the brilliant technical pioneer in the generation and distribution of AC electrical power, Charles P. Steinmetz, called the number $a +jb$ a "general" number. $\endgroup$– Richard LyonsCommented Nov 20, 2020 at 2:40
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$\begingroup$ Some dislike the term "number" because it conveys a sense of ordering or comparison that is not direct for complex quantities, and higher-order structures $\endgroup$– Laurent DuvalCommented Nov 22, 2020 at 7:08
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3 Answers
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How can a signal be imaginary?
It's not imaginary in the everyday sense of meaning something that doesn't exist.
"Real" and "imaginary" have technical meanings in signal processing, and more widely in mathematics, that are different from their everyday meanings.
In signal processing "real" denotes a signal component that is in phase with (i.e. has a 0° phase shift from) some reference signal. Imaginary denotes a signal component that is in quadrature with (i.e. has a 90° phase shift from) the same reference signal. The reference signal can come from a local oscillator. (In DSP or SDR equipment the local oscillator might be a mathematical representation of one, rather than an analog circuit.) In a receiver we might seek to synchronise the reference signal to the received signal.
In signal processing equipment we also refer to real and imaginary as I and Q for in-phase and quadrature, respectively. Some equipment has sections which are divided into separate I and Q channels. You can measure the signals flowing through a Q (imaginary) channel, implemented in hardware, and find they are just as real (in the everyday sense) as the signals flowing through the I (real) channel.
The term "imaginary" was originally coined in the 17th century by René Descartes as a derogatory term and has been confusing students ever since.
For why $e^{j \omega t} = \cos \omega t + j \sin \omega t $ see Euler's formula. Mathematicians use $i$ to represent the imaginary unit. Electrical engineers use $j$ to represent the imaginary unit because we use $i$ to represent (instantaneous) current. You can use either representation but please choose one and stick with it.
(This question was originally posted in the amateur radio SE group. This answer was drafted as a reply there and leans towards radio usage.)
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$\begingroup$ No no no! That's only a very narrow interpretation of complex signals that arise in communication systems (as in phase, I, and quadrature phase, Q, components). DSP handles complex numbers just as ordinary mathematic does.For example the Fourier transform can inherently produce complex valued signals as well; giving both the magnitude and the phase of the associated sinusoidal at the analysed frequency. So the concept of in-phase & quadrature-phase components to describe complex valued quantities is not necessary, and may even be misleading. $\endgroup$– Fat32Commented Nov 19, 2020 at 10:51
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1$\begingroup$ I prefer being a stickler about I/Q modulation as opposed to complex signals -- in that I/Q modulation very fortuitously happens to follow the same rules of arithmetic as complex numbers, but each channel is as real as can be. $\endgroup$ Commented Nov 19, 2020 at 20:52
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$\begingroup$ @Fat32 In "On Governors" (1864, James Clerk Maxwell), Maxwell refers to polynomial roots of the form $a + b\sqrt{-1}$ as composed of "possible" and "impossible" parts. $\endgroup$ Commented Nov 19, 2020 at 20:53
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2$\begingroup$ @Fat32 I'm not trying to create a complete write-up of complex signal processing; I'm just trying to illustrate an answer to OP's specific question "How can a signal be imaginary?" The name is confusing, even Gauss thought so, and I hope I've been able to explain the difference between the everyday and mathematical meanings of imaginary. $\endgroup$ Commented Nov 19, 2020 at 22:40
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$\begingroup$ @Fat32 If you want to provide an answer with a different emphasis please feel free; I'll read it with interest. $\endgroup$ Commented May 3, 2022 at 11:13
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Complex expressions are a quite common representation for several signals having two components possessing special properties.
First, it is common in NMR (Nuclear magnetic resonance) or MRI (magnetic resonance imaging), where the signal can be detected by receiver coils sensitive to magnetic flux in two orthogonal directions (cf. Real vs Imaginary Signals: What is the difference between real and imaginary signals?: How can one be more real than another?). Polarized waves, acquired by multidirectional sensors, also lend themselves well to a complex representation.
Second, creating an analytical signal from a real one combined with a quadrature part is a quite useful signal preprocessing.
They are convenient representation. I am not sure that a signal is complex per se, however, complex quantities definitely are the backbones of an apparent "real world".
answered Nov 17, 2020 at 22:35
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I would appreciate feedback, however, I would probably think of this in two steps to make it easier. First of all, there aren't any "imaginary" signals...if we think of voltage, in the end we are only concerned about the positive or negative, real value of the voltage, which in alternating current most typically has a sinodial form, let's assume it's some function U(t) = cos (wt).
In many instances it is easier to work in the exponential form as opposed to the trigonometric form, so we like transforming our signal into the exponential realm, which happens to require the complex representation. As per the Euler formula we know that exp(jwt) = cos (wt) + j sin(wt). That being said, we still have to add this imaginary sin component in order to transfer this into the exponential form. In order to again get our voltage, we'd only be concerned about the real component of this, hence U(t) = Re{exp(jwt)} = Re {cos (wt) + j sin (wt)} = cos(wt).
If I now were to take the signal j * exp(jwt), so an imaginary signal we'd get: j * exp(jwt) = j cos(wt) + j * j * sin (wt) = j cos(wt) - sin (wt). However, I am again only concerned about the real component of this which is: Re{j cos(wt) - sin(w)} = - sin(wt) = cos (wt + 90°).
As such the imaginary signal represents it's real counterpart shifted by 90 degrees.
