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RF/Microwave?? # Frequency domain tutorial: Understanding spectral components (Part II)

Posted: 04 Feb 2009 ?? ? Print Version ?

Keywords:components spectral? signal quadrature? tutorial domain?

Discrete frequency-axis notation
In the world of DSP, for convenience, frequency-domain drawings are often labeled in hertz using the fs sampling rate. This convention is best explained with a couple of examples; the first of which is when we perform spectrum analysis (using the FFT) of, say, a real time-domain audio sequence obtained at an fs = 11.025kHz rate. We could plot our spectral magnitude results using either frequency-axis labeling convention shown in Figure 11. If we later discovered that the sample rate was actually fs = 22.05kHz, we would not have to repeat our spectral analysis nor redraw our spectral plots because the frequency axis is referenced to fs. Figure 11: Example spectral magnitude plots; (a) 0Hz on the left, (b) 0Hz in the center.

Another example of labeling frequency-domain plots using hertz is in describing digital filters. A five-point moving average digital filter has the frequency magnitude response shown in Figure 12a. That frequency response curve is the same whether the filter is used in an fs = 40MSps digital communications system or in an fs = 8KSps telephone system. Figure 12: Frequency magnitude response of a 5-point moving average digital filter.

DSP authors have several other choices in labeling the frequency-axis of their frequency-domain plots. For example, the cyclic frequency (Hz) labels in Figures 11 and 12 can be converted to radians/second. We do so by replacing fs with ωs, where the signal data sample rate is Equation 6

with ωs measured in radians/second as shown in Figure 12b.

Sometimes, to make the notation more concise, DSP purists assign fs a value of one which leads to the notation that ωs = 2π. Thus, in their DSP books you'll see frequency-domain plots like Figure 13a where the frequency-axis is a normalized angle with - fs/2 replaced with - π, fs/2 replaced with π. The justification for doing so goes something like this: Let's represent a sinewave, whose frequency is fHz, by x(t) = sin(2πfot). Discrete-time samples of x(t) are: Equation 7

where the integer n sequence is the sample number (often called the "index") of x(n). With the factors 2πf having the dimension of radians/second, and ts having the dimension seconds/sample, the resultant angle in Equation 7 has the dimension of radians/sample. If we replace Equation 7's ts with 1/fs, the discrete sinusoidal samples can be represented by: Equation 8

where θ is what I call a "normalized discrete-signal frequency." If we assume |fo| ≤ fs/2 (satisfying Nyquist), the normalized discrete-signal frequency θ is in the range of –π to +π measured in radians/sample. This definition is why some authors like to say, "For continuous signals, frequency is measured in radians/second. For discrete signals frequency is measured in radians/sample." Redrawing the filter response from Figure 12b, we illustrate the normalized discrete-signal frequency-axis representation in Figure 13a.

Just so you know that I'm not making all of this up, Figure 13b shows how a Matlab built-in plotting function uses the radians/sample frequency notation. Figure 13: Filter response plots using the normalized discrete-signal frequency notation of radians/sample.

If you've spent your technical career thinking about frequency measured in cycles/second (Hz), the frequency-axis labeling in Figure 13 might seem very odd. However, it's not so strange. Consider the discrete sinewave in Figure 14a, whose sample values repeat every 12 samples. It takes 12 samples to complete one cycle (360°) of oscillation. Likewise, we can say it takes 6 samples to complete one radian (180°) of oscillation. From that last statement, we declare the discrete-signal frequency of the sinewave to be one sixth radians/sample. A spectral plot of the sinewave is shown in Figure 14b. Figure 14: A discrete sinewave, (a) time-domain samples, (b) frequency-domain samples.

To consolidate our thoughts, we list various frequency-axis notations in the table. The third column shows the frequency range of analysis when using the FFT. Table: Various frequency-axis notation.

It often takes a DSP novice some time to become comfortable with these various frequency-axis notations. Fortunately, commercial signal processing software packages like LabVIEW, Mathcad and Matlab allow us to conveniently label our frequency-domain plots in good ol' hertz.

This is the second part of a two-part series.

- Richard Lyons
Besser Associates

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