How can we interpret the negative side of a double-sided frequency spectrum

We know that a frequency spectrum includes both positive and negative frequencies. However, how can we explain the negative frequency physically?

For example, in a wireless communication system, suppose the carrier frequency is 2.4GHz. The power carried by this frequency is, say 10uW. In a double-sided spectrum, this carrier frequency can be represented by two tones, one at f = 2.4GHz, another at f = -2.4GHz. Each of these tones carries a power of 5uW.

What I understand is that, the rotations of these two tones are in opposite directions. However, in the carrier signal, what is rotating? Is it related to the propagation of the EM waves?

Could anyone offer some insights?

Thanks.

• Each frequency has a real and imaginary part. If you sample and only record the real part, you recorded only half the information per sample and so your positive frequencies are mirrored. If you sample and record both real and imaginary part (e.g. coherent receiver), then the negative frequencies can be different from the positive. Sep 28, 2020 at 23:30
• Can you explain how you interpret te positive side of a frequency spectrum, and make clear why that interpretarion is contradicting with a negative spectrum? Sep 28, 2020 at 23:57
• If you graph cos (2*pi*2.4GHzt) and compare it with the graph of cos (-2*pi*2.4GHzt) you will see that positive and negative frequencies are the same thing (assuming you start from t=0). This makes sense. Any time you have a positive frequency (increasing phase) you also have a negative frequency (decreasing phase). They are mathematically identical. If you use sine or start at a time when the phase is not equal, then you will still have positive and negative frequencies, but they will be offset in phase with respect to each other. Sep 29, 2020 at 0:49
• In reality, there are no "negative frequencies" . They are nothing else than the result of EULERs method for writing down angular functions. More than that, there are no "frequencies" at all...there exist only signals which have some properties like frequency, amplitude, phase shift,....
– LvW
Sep 29, 2020 at 8:35
• Possible duplicate of: electronics.stackexchange.com/q/102528/6494 Oct 1, 2020 at 17:53

It is useful to think of observable (real-valued) waves as the sum of two complex waves:

• it doesn't require a nonlinear operation to drop the imaginary part, like $$\\cos x=\Re(e^{ix})\$$ does.
• it doesn't require an (arbitrary) definition whether an observed sine wave has increasing or decreasing phase (because $$\\sin x=-\sin -x\$$).
• it explains mixing behaviour

The latter is the most important point for practical purposes: if you mix two sine waves by multiplying them, you end up with two sines, one with the sum of the frequencies, one with the difference, and a phase shift:

$$(\sin x)(\sin y)=\frac{\cos(x-y)-\cos(x+y)}{2}$$

This makes a lot more sense if you substitute

$$\begin{eqnarray} \sin u&=&\frac{e^{-iu}-e^{iu}}{2}i\\ \cos u&=&\frac{e^{-iu}+e^{iu}}{2} \end{eqnarray}$$

to get

$$\begin{eqnarray} (\frac{e^{ix}-e^{-ix}}{2}i)(\frac{e^{iy}-e^{-iy}}{2}i)&=&-\frac{(e^{ix}-e^{-ix})(e^{iy}-e^{-iy})}{4}\\ &=&-\frac{e^{ix}e^{iy}-e^{-ix}e^{iy}-e^{ix}e^{-iy}+e^{-ix}e^{-iy}}{4}\\ &=&-\frac{e^{i(x+y)}-e^{i(-x+y)}-e^{i(x-y)}+e^{i(-x-y)}}{4}\\ &=&-\frac{e^{i(x+y)}-e^{-i(x-y)}-e^{i(x-y)}+e^{-i(x+y)}}{4}\\ &=&-\frac{e^{i(x+y)}+e^{-i(x+y)}-e^{-i(x-y)}-e^{i(x-y)}}{4}\\ &=&-\frac{(e^{i(x+y)}+e^{-i(x+y)})-(e^{-i(x-y)}+e^{i(x-y)})}{4}\\ &=&-\frac{\cos(x+y)-\cos(x-y)}{2}\\ &=&\frac{\cos(x-y)-\cos(x+y)}{2} \end{eqnarray}$$

That way it is also way more intuitive how the spectrum of the multiplication of two signals is the convolution of the signal's spectra: the input signals have two peaks each, and the output signal has four.

This system also remains consistent when you operate on complex-valued signals, e.g. digital baseband, where it is possible to mix with a complex-valued sine and get a frequency shift without moving half the signal's energy to a mirror image.

The mixing behaviour is observable in the real world, so a model that adequately explains it is useful.

• nice answer and +1. But... sin(x) = sin(-x)...? Did you forget something? Sep 29, 2020 at 14:49
• @SredniVashtar, oops, indeed. Sep 29, 2020 at 14:50
1. The carrier frequency is either 2.4GHz or something else, but not both. Moreover, it makes no sense to speak of the negative frequency in the context of EM radiation frequency.

2. Your representation of a carrier wave by "two tones" is not valid (item 1) and so your consideration of power splitting between waves not applicable. As a side note: you cannot separate the total power at the receiver antenna into the contributions of individual waves without taking into account the interference.

3. Because your "two tones" are not defined, we cannot discuss their "rotation". Still, talking about rotation: in a circularly polarized electromagnetic wave, the field vectors rotate. This rotation can be either clockwise or counter clockwise w.r.t. the wave vector direction (propagation direction) and, in principle, it can be used to carry data (sort of "polarity shift keying" data coding), but has nothing to do with negative frequency.

The spectrum of electromagnetic signals and the electromagnetic radiation spectrum are different things. To understand why the signal spectrum contains both positive and negative frequency values, concentrate on the EE discipline. Learn phasors, transfer functions, poles/zeroes, constellation diagrams, modulation/demodulation etc.

...let me offer my own redneck point of view. I'm not a graduated EE techie and the more arcane parts of the math are lost on me. I'm trying to use the tools like I would use a shovel (or a third-party programming library, whom I trust to do its job).

I've met this phenomenon of negative spectra while trying to put KissFFT to work as part of my proggie called the RtlSDR Skyline.

FFT is a transform that works upon a buffer of complex samples. The number of samples, AKA the Window, should preferably be a power of two. Such as, 2048 samples. Each sample is a complex number, encoding two orthogonal components of the signal, or amplitude and phase if you prefer to look at it that way. So the transform takes a buffer of CPX samplex of the time-domain signal, and returns a buffer of the same size, only containing complex samples of frequency spectrum (spectral lines with amplitude and phase). Within the context of some "assumptions out of band" about infinities etc, the input buffer and the output buffer carry the same amount of information. The number of time-domain samples is equal to the number of frequency-domain samples. And, apologies for skipping some math, this whole algorithm implies, that the frequency data buffer contains twice the bandwidth, that you would normally expect by applying the popular Nyquist limit: i.e., that the maximum frequency component represented in the time-domain signal is at half the sampling rate. I.e., looking at the spectral output buffer, the lower half is what you would expect, and the upper half are mirror frequencies, a result of aliasing, wrapped around the nyquist frequency. Or, maybe there's a better way to look at this "aliased half": consider them the negative frequencies :-) = wrapped around the DC bin. And, they are just the same worth as the positive half of the spectrum. An important point: does this mean, that the two half-spectra are necessarily mirror images of each other? No, they are not! :-)

Let me quote from the KissFFT docs: "Note: frequency-domain data is stored from dc up to 2pi. so cx_out[0] is the dc bin of the FFT and cx_out[nfft/2] is the Nyquist bin (if exists)". In my program, the first thing I do with the spectral data is: I swap the two halves in the buffer, so that I get a continuous spectrum of -(bandwidth/2) .. +(bandwidth/2)` , with the DC bin in the middle.

At this point, let me cut to a side plot:

Why complex numbers? The explanation in pure physics might be, that a vibrating system, especially if this is a resonant sinusoidal vibration, actually contains two forms of energy, that keep rotating, circulating, spilling from one form to the other and back again. A metastable system containing a constant amount of energy. Kinetic energy to positional potential energy. Current to Voltage and vice versa. Electric vs. magnetic field. Or some such. (In an environment with some characteristic impedance, which is a condition for physical "work" and "power" to arise - but let's not tighten the math too far.)

In a purely voltage-coupled signal path, the complex number may carry information about phase of the "pure voltage" signal - which raises the question how the phase was arrived at, if there's no current associated :-) What does the phase encode?

Cutting back to my FFT-based band-scanning software:

The complex I+Q samples that you get out of the demodulator, those are a product of mixing. At the input of the quadrature demodulator, you have a narrow-band intermediate frequency signal, which itself is already a product of down-mix/downconversion from some carrier frequency. The quadrature demodulator takes the arbitrary input signal, whose spectrum is centered around the nominal IF frequency (shaped by a passive band-pass filter that "selects" a band of frequencies around the nominal IF) and multiplies that signal by a pure sinewave of the IF frequency. The quadrature demodulator actually uses two reference sinewaves, shifted 90* (or sine and cosine) - so you get two multiplication products, called I and Q. These are still time-domain samples, but each sampled "timeslot" contains a complex number.

Now... suppose that the signal coming down the IF path is actually a pure sinewave exactly at the IF frequency, identical to the local oscillator. In that case, on the output of the demodulator, you get a constant complex number, containing an amplitude of the received IF signal, and a phasor = phase angle of the received sinewave, as compared to the local oscillator's reference signal. The phase angle is decoded using the two reference waves, phase-shifted by 90*.

Next step: suppose that the sinewave, coming down the IF path, is a little slower than the local oscillator. What happens? The complex phasor, coming out of the demodulator in the form of I+Q samples, will rotate.

Next: suppose the sinewave, coming down the IF path, is a little faster than the local osc. What happens? Again the complex I+Q phasor will rotate, but in an opposite direction.

So: here we have a time series of complex I+Q samples (phasors sampled in discrete time instants). What happens if we run this through FFT? That's right, for the three examples above, we get a DC component, a negative and a positive frequency line in the FFT output.

Note: you need to consider the broader superhet system at hand. The "positive and negative" is relative to the IF channel's center frequency. As soon as I plot the partial band just decoded into the overview spectrum, it will sit on the spectral scale at absolute frequencies that are definitely positive (after I've added the "tuned carrier center frequency").

So if I tune my RTLSDR dongle into say 100 MHz carrier frequency, and the sampling rate is 2 MSps, as a net result on the screen I can get a channel worth of 2 MHz of spectrum, centered around 100 MHz. That's 99 to 101 MHz. If there's a radio transmitter at 99,7 MHz, I should see a peak around -300 kHz in my raw FFT output.

You may ask: that's all fine, but how does FFT come up with a spectrum of a real-world IF signal, that looks mostly like noise, centered around your local oscillator frequency? The demodulator will keep spewing I+Q samples with random absolute values and phases. What's the point? You cannot tell a frequency out of a single random sample, or a sequence of very random samples, right? Well, wrong :-) A single sample indeed doesn't contain frequency information. But a time series, that's a whole different matter. For some nice results appealing to the human eye, such as a peak in the otherwise noisy spectrum, the time series of CPX samples must contain some repetitive pattern. It may not be well visible to the naked eye in the time domain waveform, but the time series buffer is a statistical dataset, over which you can decode individual frequency components by a simple statistical operation called the convolution. You multiply samples of a time series (time-domain signal) by samples of a "probe waveform" and accumulate the per-sample products. If the accumulated result is large, it means that your probe wavelet "rings a bell" :-)

("Ooohhh waiwaiwait, how come that modems or TV transponders produce such a nicely shaped brick of noise? Allegedly modulated by QPSK/QAM/OFDM. How does that carry some clean data?" Answer: the signal is scrambled by a pseudo-random sequence to be noisy, or at least coded by 10/8 or some such to be free of a DC component, forward error correction is added, on duplex channels the media can be trained and the signal pre-equalized, etc. That's where my brain is already squirting out my ears = stop asking.)

So what the Discrete Fourier Transform does: it meticulously convolves the time-series buffer with every "frequency bin" of the output, and places the accumulated product into the output bin. Fast Fourier Transform is a version of DFT that's heavily optimized for compute efficiency - an explanation is outside of scope here (and beyond my grasp anyway).

So... how do negative frequencies come out of this meticulous convolution algorithm? My take: apparently the "frequency probe waveform" per spectral bin is a complex time series too, and can be made to rotate clockwise or counter-clockwise? And, I seem to recall something about complex conjugates and correlation between the positive and negative spectrum being used to calculate (and post-compensate) I/Q imbalance, but that's where I'm getting lost :-)

Note that the raw spectral samples coming out of FFT are not only positive/negative (explained above) but on top of that, they are complex too... which in this case means, that apart from amplitude of each particular spectral line (FFT output bin), we also know its phase angle within the FFT window. For practical purposes in a simple spectrum analyzer, the phase is not really interesting - therefore, for spectral display on the screen, the only sensible preprocessing operation is to take the absolute value (amplitude) of each complex frequency bin = turn the complex phasor into a scalar amplitude.

2,4GHz sinusoidal AC voltage: Its properly calculated Fourier spectrum in modern communication theory texts really contains components at +2,4GHz and -2,4GHz. But that's the spectrum, only an image of the signal, not the signal itself. It's useless to start to dream there's somewhere lurking also a voltage which has frequency -2,4GHz.

You can smudge things more. I guess you have already noticed that a distorted sinewave can be divided for ex. with precise filters to harmonic components and if those components are summed back together the original signal is reconstructed.

Let's imagine we have distorted sinewave named X and its frequency = 1kHz. Let's imagine X is divided to harmonic components with a precise filter bank. It outputs voltages which have frequencies 0Hz(=DC), 1kHz, 2kHz. 3kHz, etc... A summing mixer produces the original X if the input signals are those harmonic components.

No negative frequencies are needed.It looks like Mr. Fourier got too much sun on his head when he wandered in Egypt with Mr. Napoleon Bonaparte.

BUT: Actually they are not needed. They are a later innovation than Fourier's original work (see NOTE1). Today in communication theory we do not consider signals as a sum of pure continuous sinewaves. We consider them to be a sum of complex exponentials. That makes possible to have also complex valued signals and do calculations with their spectrums.

Of course real valued signals in that presentation will have equally strong spectrum components at positive and negative components. That cancels the imaginary part of the signal as you should easily see if you watch Euler's identity exp(jAt)=cos(At)+jsin(At).

In Fourier's original spectrum presentation a signal is a sum of continuous sine signals which have certain positive frequencies, amplitudes and phase angles. No complex number valued signals was attempted to divide to frequency components.

So, what in the hell are those complex valued signals? Believe or not, in signal processing we can have the real part and the imaginary part as separate voltages or as well we can have 2 series of samples in DSP. I have had in my hands a radar receiver where the analog signal processing circuit generated complex valued intermediate results (I and Q channels, I guess you'll see them numerous times). That kind of processing simplified radically the circuit because frequency shifting with mixers was no more ambiquous, so called image frequency ambiquity didn't exist. In addition synchronous detection and pulse compression happened perfectly with no worries of phase shifts.

Finally all kind of formulas in elementary communication theory become much simpler if we calculate spectrums with modern formula where the basic periodic signal is the complex exponential.

The only drawback is that the spectrum of a real signal in complex exponential base system has equally strong components at positive and negative frequencies and that can cause some confusion. But as said, that's the spectrum, not the signal itself. A real number valued signal could as well be presented as a sum of positive frequency real valued sinewaves.

NOTE1: The idea of presenting a math function as a sum of continuous sinewaves was not Fourier's invention. It can be found in earlier works. Fourier became famous because he found a systematic method to useit in solving practical problems. Electronics was still non-existent, but F. worked with heat transfer and mechanical vibrations.