With this thought in mind, we could draw the spectral magnitude (ignoring any phase information) of a continuous 400 Hz sinusoid as shown in Figure 9(a) showing the inherent spectral symmetry about zero Hz when we represent real signal spectra with complex exponentials. By 'real signal' we mean an x(t) signal having a non-zero real part but whose imaginary part is always zero. (Our convention is to treat all signals as complex and to think of real signals as a special case of complex signals.) Figure 9(a) is another graphical representation of Euler's identity in Eq. (3).

Figure 9. The spectral magnitude plot of (a) a 400 Hz continuous sinusoid, and (b) a discrete sequence of a 400 Hz sinusoid sampled at a 2 kHz sample rate.

If we apply our convention of 'spectral replications due to periodic sampling', we can illustrate the spectral magnitude of discrete samples of a 400 Hz sinusoid, sampled at an f_s = 2 kHz sampling rate, as that in Figure 9(b). And so there you are. Figure 9(b) is typical of the spectral magnitude representations used in the DSP literature. It combines the spectral replications (centered about integer multiples of f_s) due to periodic sampling as well as the use of negative frequency components resulting from representing real signals in complex notation. (Whew!)

To review the spectrum of another discrete sequence, Figure 10(a) shows the spectral magnitude of a continuous x(t) signal having four components in the range of 100 Hz to 700 Hz where dark and light squares distinguish the positive and negative-frequency spectral components. Figure 10(b) shows the spectral replication for a discrete x(n) sequence that's x(t) sampled at 2 kHz. The sole purpose of this article is to show the meaning, relevance, and validity of Figure 10(b) in representing the spectrum of discrete samples of a real sinusoid in the complex-valued world of DSP. This figure reminds us of the following important properties: continuous real signals have spectral symmetry about 0 Hz; discrete real signals have spectral symmetry about 0 Hz and ±f_s/2 Hz.

Figure 10. Spectrum of a signal with four components in the range of 100 Hz to 700 Hz. (a) Spectral magnitude of the continuous signal. (b) Spectrum of a sampled x(n) sequence when f_s = 2 kHz, and (c) spectrum of the x'(n) sequence when f_s = 1.3 kHz.

Figure 10 illustrates why the Nyquist Criterion for lowpass signals—signals whose spectral components are centered about zero Hz—states that the f_s sampling rate must be equal to or greater than twice the highest spectral component of x(t). Because x(t)'s highest spectral component is 700 Hz, the f_s sample rate must be no less than 1.4 kHz. If f_s were 1.3 kHz as in Figure 10(c), the centers of the spectral replications would be too close together and spectral overlap would occur. We see that the spectrum in the range of –1 kHz to +1 kHz in Figure 10(c) does not correctly represent the original spectrum in Figure 10(a). This unfortunate situation is typically called aliasing, and it results in x'(n) sample values that contain amplitude errors. For real-world, information carrying, signals there is no way to correct for those errors.

For clarity, let's describe this situation using different words. Given the proper sampling shown in Figure 10(b), we could apply the x(n) samples to a digital-to-analog converter, followed by high-performance analog filtering, and exactly regenerate (reconstruct) the original analog x(t) signal. With the improper sampling in Figure 10(c), there is no way to generate the original analog x(t) signal using the corrupted x'(n) samples.

In Figure 10(c) we can see that the spectral overlap is centered about f_s/2 and that particular frequency is important enough to have its own name; it's sometimes called the folding frequency, but more often it's called the Nyquist frequency. We can make the following very important statement relating continuous and discrete signals, "Only continuous frequency components as high as the Nyquist frequency (f_s/2) can be unambiguously represented by a discrete sequence obtained at an f_s sampling rate." Figure 10(c) also reminds us of another fundamental connection between the worlds of continuous and discrete signals. All of the continuous x(t) spectral energy shows up in the discrete x'(n) sequence's spectral frequency range of –f_s/2 to +f_s/2.

The purpose for showing replicated spectra as we did in Figure 10 is not to cause complication or confusion, but to provide a straightforward explanation for the effects of overlapped spectra due to aliasing. (Drawing replicated spectra is also useful in illustrating the spectral translation that takes place in bandpass sampling, and describing the result of frequency translation operations such digital down-conversion.) With that said, we conclude this article with an explanation of the various, and sometimes puzzling, notations used for frequency-axis labeling in the DSP literature. Don't touch that dial.