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This series is drawn from the course "DSP Made Simple for Engineers." For more information, see Besser Associates. This article is also available as a PDF
Part 1 discusses the ambiguities of discrete signals.
A brief introduction to quadrature signals
Let's now focus on describing a quadrature signal, having a real and an imaginary part, that is a function time. To do so we must remember that, as the great mathematician Karl Gauss first recommended, a single complex number can be represented by a point on the two-dimensional complex plane. Such a plane has two axes (real and imaginary) that are orthogonal to each other, meaning there is a 90° difference in the axes' orientations. Consider a complex number whose magnitude is one, and whose phase angle increases with time. That complex number is the ej2πfot point shown on the complex plane in Figure 4. (Here the 2πfo term is frequency in radians/second corresponding to a frequency of fo cycles/second where fo is measured in Hz.) As time t increases the complex number's phase angle φ = 2πfot increases and our number orbits the origin of the complex plane in a counterclockwise direction. Figure 4 shows that number, represented by the solid dot, frozen at some arbitrary instant in time. (That rotating ej2πfot complex number goes by two names in the DSP literature; it's often called a "complex exponential", and it's also referred to as a "quadrature signal.") If, say, the frequency fo = 2 Hz then the solid dot would rotate around the circle two times, or two cycles, per second.
Figure 4. A snapshot, in time, of two complex numbers whose exponents, and thus their phase angles, change with time.
Because complex numbers can be represented in both polar and rectangular notation, we can represent our polar ej2πfot quadrature signal (using one of Leonhard Euler's identities) in rectangular form as:
ej2πfot = cos(2πfot) + jsin(2πfot). (2)
Equation (2) tells us that as ej2πfot
rotates around the origin its real part, its East-West distance from the origin, varies as a cosine wave. The complex exponential's imaginary part, the North-South distance from the origin, varies as a sinewave. (Understanding the nature of a sinusoidal quadrature signal is no more difficult than reading a road map.) The attributes of our two-dimensional ej2πfot
complex exponential are best illustrated with a three dimensional time-domain plot as in Figure 5. Notice how the ej2πfot
signal spirals so beautifully along the time axis with its real part being a cosine wave and its imaginary part being a sinewave. At time t = 0 the signal has a value of 1 + j 0 as we would expect. (Equation (2) allows us to represent a single complex exponential as the orthogonal sum of real cosine and real sine functions.)
Figure 5. The value of the ej2πfot complex exponential signal.
That ej2πfot
signal is not just mathematical mumbo jumbo! We can physically generate an ej2πfot
signal and transmit it to a laboratory down the hall. All we need is two equal-amplitude sinusoidal signal generators, set to the same frequency fo. (However, somehow we have to synchronize those two hardware generators so that their relative phase shift is fixed at 90°. Their outputs need to be orthogonal.) Next we connect coax cables to the generators' output connectors and run those two cables, labeled 'Cosine' for our cosine signal and 'Sine' for our sinewave signal, down the hall to their destination. In the other lab, if the continuous real signals were connected to the horizontal and vertical input channels of an oscilloscope, as in Figure 6, we'd see a bright spot rotating counterclockwise in a circle on the scope's display. (Remembering, of course, to set the scope's Horizontal Sweep control to the 'External' position.)
Figure 6. Quadrature ej2πfot signal oscilloscope display.
Pop quiz: What would be seen on the scope's display if the cables were mislabeled and the two real signals were inadvertently swapped? If you said we'd see another circle orbiting in a clockwise direction, pat yourself on the back because you'd be correct.
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