This oscilloscope example helps us answer the important question, "When we work with quadrature signals, how is the j‑operator implemented in hardware?" The answer is that the j‑operator is implemented by how we treat the two real signals relative to each other. We have to treat them orthogonally such that the cosine signal represents the Real (East-West) value, and the sinewave signal represents the Imaginary (North-South) value. So in our oscilloscope example the j‑operator is implemented merely by how the connections are made to the scope, and the result is a two-dimensional quadrature signal represented by the instantaneous position of the dot on the scope's display.

By the way, if we control the instantaneous phase of the e^j2πf_ot signal based on some bipolar binary data (+1 and –1), a person on the other lab could measure that phase at certain instants in time and extract that binary data. Many digital communications systems operate on this principle. OK, back to business. At this point you may ask, "Where does the idea of negative frequency come in here?" Well, there's a 'Negative Frequency' signpost up ahead and we're now ready to answer that question.

Don't be negative about negative frequency
The notion of negative frequency is often troubling to engineers who've spent so much time examining the spectra displayed on analog spectrum analyzers. Some engineers think of frequency, by its very nature, as something that cannot be negative. Such as, say, starting your car and driving minus ten miles. Well, we can give negative frequency a solid physical meaning by defining it properly in the context of complex, or quadrature, signals. Let's do that now.

Returning to Figure 4, we can also think of another complex exponential e^–j2πf_ot, the white dot, orbiting in a clockwise direction because its phase angle φ = –2πf_ot becomes more negative as time increases. Again, if the frequency f_o = 2 Hz then the white dot would rotate around the circle two times, or two cycles, per second in the clockwise direction. By definition, we call that rotational frequency minus two cycles per second. Those two complex exponentials in Figure 4 are of great interests to us because of what is obtained when they're summed algebraically. For example, what is the sum of the positive-frequency counterclockwise rotating e^j2πf_ot and the negative-frequency clockwise rotating e^–j2πf_ot when we add their real and imaginary parts separately? That's right. The sum is a oscillating function whose imaginary part is always zero. That real-only sum is a cosine wave whose peak amplitude is 2. If the magnitudes of the complex exponentials in Figure 4 had been 0.5, instead of 1, they would graphically depict another important Euler identity:

cos(2πf_ot) = e^j2πf_ot/2 + e^–j2πf_ot/2 .    (3)

Equation (3) allows us to represent a real cosine wave as the sum of positive-frequency and negative-frequency complex exponentials. By our definitions, a positive-frequency complex exponential's exponent is positive, and a negative-frequency complex exponential has a negative exponent.

Another Euler identity, Eq. (4), gives the relationship of a real sinewave as the sum of positive-frequency and negative-frequency complex exponentials.

sin(2πf_ot) = j(e^j2πf_ot/2) – j(e^–j2πf_ot/2).    (4)

Those j‑operators in Eq. (4) merely describe the relative phase of the complex exponentials at time t = 0 as illustrated in Figure 7.

Figure 7. The two complex exponentials, at time t = 0, that comprise a sinewave.

At time t = 0, Eq. (4) becomes:

sin(2πf_ot) | _t=0 = j (e^–0/2) – j (e⁰/2) = j/2 – j/2 = 0    (5)

complying with our knowledge that a sinewave's amplitude is zero at time t = 0. Don't worry if these concepts of the j‑operator and complex exponentials seem a little perplexing at this point. You'll get used to them. (Even the great Karl Gauss struggled with these ideas at first. He called the j‑operator the "shadow of shadows".)

OK, let's not forget where we're going here. Our ultimate goal is understand the nature of the spectral diagrams used in DSP. In doing so we had to define the notion of negative frequency and that definition is inherent in the complex-valued (real and imaginary) representation we use for discrete spectra in DSP. Unlike the amplitude-only results seen when you use an analog spectrum analyzer, in the world of DSP our spectrum analysis provides complex-valued results. That is, discrete spectra show the relative phase shifts between spectral components.

Let's look at the complex spectra of a few simple sinusoids, from the viewpoint of Euler's identities, as shown in Figure 8. The time-domain waveform and the complex spectra of a sinewave defined by sin(2πf_ot) is shown in Figure 8(a). Shifting that sinewave in time by 90° gives us a cosine wave shown in Figure 8(b). Another shift in time by φ° results in a arbitrary-phase cosine wave in Figure 8(c).

Remember now, the positive and negative-frequency spectral components of the sinewave rotated counterclockwise and clockwise, respectively, by 90° in going from Figure 8(a) to Figure 8(b). If those cosine wave spectral components continued their rotation by φ° we'd have the situation shown in Figure 8(c). We show these three-dimensional frequency-domain spectra, replete with phase information, because in the world of DSP we're often interested in spectral phase relationships. We use the FFT algorithm to measure spectral magnitude and phase the way an analog engineer uses a network (vector) analyzer. (In case you hadn't noticed, Figure 8 illustrates a very important signal processing principle. A time-domain shift of a time-periodic signal results only in phase shifts in the frequency domain, spectral magnitudes do not change.)

Figure 8. Complex frequency domain representation of three sinusoids.

The top portion of Figure 8 illustrates Eq. (3) and the center portion is a graphical description of Eq. (4). Thankfully, we've almost reached our goal! Figure 8 reminds us that one legitimate way to show the spectrum of a real cosine wave is to include both positive and negative-frequency spectral components.