This chapter treats frequency-domain analysis by developing the discrete-time Fourier transform (DTFT) and the spectral representation of discrete-time signals. The development, properties, and applications of the DTFT closely parallel those for the continuous-time Fourier transform. In fact, the DTFT and the CTFT are closely related, a fact that we can use to our advantage in many applications. Discrete-time Fourier analysis provides an important tool for the study of discrete-time signals and systems. A frequency-domain perspective is particularly useful to better understand the digital processing of analog signals and digital resampling techniques. In preparation for the next chapter, this chapter concludes by generalizing the DTFT to the z-transform.

The Discrete-Time Fourier Transform

The continuous-time Fourier transform (CTFT), sometimes simply called the Fourier transform (FT), is a tool to represent an aperiodic continuous-time signal in terms of its frequency components, thereby providing a spectral representation of the signal. We now develop a similar tool, the discrete-time Fourier transform (DTFT), to represent an aperiodic discrete-time signal in terms of its frequency components, which also leads to a spectral representation of the signal. The principal difference between the two types of transforms is that the former represents signals with continuous-time sinusoids or exponentials, while the latter uses discrete-time sinusoids or exponentials.

One possible way to develop the continuous-time Fourier transform is to start with the Fourier series representation of periodic signals and then, letting the period go to infinity, extend the results to aperiodic signals.

Since its emergence as a field of importance in the 1970s, digital signal processing (DSP) has grown in exponential lockstep with advances in digital hardware. Today's digital age requires that under-graduate students master material that was, until recently, taught primarily at the graduate level. Many DSP textbooks remain rooted in this graduate-level foundation and cover an exhaustive (and exhausting!) number of topics. This book provides an alternative. Rather than cover the broadest range of topics possible, we instead emphasize a narrower set of core digital signal processing concepts. Rather than rely solely on mathematics, derivations, and proofs, we instead balance necessary mathematics with a physical appreciation of subjects through heuristic reasoning, careful examples, metaphors, analogies, and creative explanations. Throughout, our underlying goal is to make digital signal processing as accessible as possible and to foster an intuitive understanding of the material.

Practical DSP requires hybrid systems that include both discrete-time and continuous-time components. Thus, it is somewhat curious that most DSP textbooks focus almost exclusively on discrete-time signals and systems. This book takes a more holistic approach and begins with a review of continuous-time signals and systems, frequency response, and filtering. This material, while likely familiar to most readers, sets the stage for sampling and reconstruction, digital filtering, and other aspects of complete digital signal processing systems. The synergistic combination of continuous-time and discrete-time perspectives leads to a deeper and more complete understanding of digital signal processing than is possible with a purely discrete-time viewpoint. A strong foundation of continuous-time concepts naturally leads to a stronger understanding of discrete-time concepts.

In this chapter, we introduce the discrete Fourier transform (DFT), which may be viewed as an economy class DTFT and is applicable when x[n] is of finite length (or made finite length by windowing). The DFT is one of the most important tools for digital signal processing, especially when we implement it using the efficient fast Fourier transform (FFT) algorithm, discussed in Sec. 9.7. The development of the FFT algorithm in the mid-sixties gave a huge impetus to the area of DSP. The DFT, using the FFT algorithm, is truly the workhorse of modern digital signal processing, and it is nearly impossible to exaggerate its importance. A solid understanding of the DFT is a must for anyone aspiring to work in the digital signal processing field. Not only does the DFT provide a frequency-domain representation of DT signals, it is also useful to numerous other tasks such as FIR filtering, spectral analysis, and solving partial differential equations.

Computation of the Direct and Inverse DTFT

As we saw in Ch. 6, frequency analysis of discrete-time signals involves determination of the discretetime Fourier transform (DTFT) and its inverse (IDTFT). The DTFT analysis equation of Eq. (6.1) yields the frequency spectrum X(Ω) from the time-domain signal x[n], and the synthesis equation of Eq. (6.2) reverses the process and constructs x[n] from X(Ω). There are, however, two difficulties in the implementation of these equations on a digital processor or computer.

MATLAB (a registered trademark of MathWorks, Inc.) is a language and interactive environment for numeric computation, algorithm development, data analysis, and data visualization. It is particularly well suited for digital signal processing applications. While MATLAB is relatively simple to use, it is a sophisticated package that can be a bit intimidating to new users. Fortunately, there are many excellent resources on MATLAB and its use, including MATLAB's own built-in documentation. This appendix provides a brief introduction to MATLAB that complements the MATLAB material found throughout this book.

Scripts and Help

When MATLAB is launched, a command window appears. Users issue commands at the command prompt (≫), and MATLAB responds, generally with the creation or modification of workspace objects. MATLAB supports different types of workspace objects, such as functions and strings, but usually objects are just data. The workspace window summarizes the names and important characteristics of currently available objects.

While users can directly input sequences of commands at the command prompt, it is generally preferable to instead use a MATLAB script file (M-file). A MATLAB script file is simply a text file (.m extension) that contains a collection of MATLAB statements. Comments are inserted by preceding text with a percent sign (%). Any text editor can be used to create an M-file, but MATLAB's built-in editor provides added functionality such as color-coded text, breakpoints, and various other features. M-files are easy to modify and facilitate rapid algorithm development. M-files are executed directly from MATLAB's editor or by typing the M-file name (without the .m extension) at the command prompt.