# Frequency domain tutorial, part 1: Dealing with ambiguity

*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
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Part 2 introduces quadrature (complex) signals, and explains the nature, and notation, of the spectral diagrams used in DSP.*

**The problem**

One of the major obstacles engineers encounter while learning digital signal processing (DSP) is understanding how spectral components are defined and illustrated in the frequency domain. When people begin to read DSP literature they encounter strange new terminology describing discrete spectra such as *folding frequency*, *aliasing*, *orthogonal*, *images*, *Nyquist*, *spectral replications*, and *negative frequency*. Typical DSP spectral diagrams initially seem peculiar because they often show negative frequency spectral components and what appear to be replicated spectral components.

Making matters worse for the inquisitive engineer, various DSP authors use different, and sometimes puzzling, notation in labeling frequency axis in their spectral plots; often the frequency dimension of hertz is not used at all in discrete spectral diagrams. For example, many university DSP textbooks actually label the discrete frequency-axis covering a range from – π to +π! The perplexing frequency-domain terminology and notation originate from a kind of frequency ambiguity inherent in discrete (sampled) systems and the fact that in DSP we sometimes describe all signals as if they were complex-valued (with real and imaginary parts). Understanding the differences between analog and discrete spectra is one of the reasons DSP has the reputation of being difficult to learn. Fortunately several books have been published that ease the engineer's burden of learning DSP.^{[1}^{–3]}

For our short journey to understanding the mathematics and notation of discrete spectra we start by discussing the frequency-domain ambiguity associated with discrete signals, and arrive at our destination of understanding the subtle aspects, the notation, and the language of the discrete frequency domain of DSP. However, as we proceed we'll make briefs stops to review complex signals, negative frequency, and discrete spectrum analysis using the fast Fourier transform (FFT).

**Frequency-domain ambiguity**

We begin by reviewing the source of one unpleasant aspect of sampled-data systems: the frequency-domain ambiguity that exists when we digitize a continuous (analog) signal *x*(*t*) with an analog-to-digital (A/D) converter as shown in Figure 1.

*Figure 1. Periodic sampling of (digitizing) a continuous signal.*

This process *samples* the continuous *x*(*t*) signal to produce the *x*(*n*) sequence of binary words that are stored in the computer for follow-on processing. (Variable '*n*' is a dimensionless integer that we use as our independent time-domain index in DSP, just as the letter '*t*' is used in continuous-time equations.) The *x*(*n*) sequence represents the voltage of *x*(*t*) at periodically spaced instants in time, and so we call the Figure 1 process "periodic sampling." We'll designate the time period between samples as *t*_{s}, measured in seconds, and define it as the reciprocal of the sampling frequency *f*_{s}, i.e., *t*_{s} = 1/*f*_{s}. In the literature of DSP the *f*_{s} sampling frequency is given the dimensions of 'samples/second', but sometimes we indicate its dimension as Hz because *f*_{s} shows up on the frequency axis of our spectral diagrams.