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The growth in the field of digital signal processing began with the simulation of continuous-time systems in the 1950s, even though the origin of the field can be traced back to 400 years when methods were developed to solve numerically problems such as interpolation and integration. During the last 40 years, there have been phenomenal advances in the theory and application of digital signal processing.
In many applications, the representation of a discrete-time signal or a system in the frequency domain is of interest. To this end, the discrete-time Fourier transform (DTFT) and the z-transform are often used. In the case of a discrete-time signal of finite length, the most widely used frequency-domain representation is the discrete Fourier transform (DFT) which results in a finitelength sequence in the frequency domain. The DFT is simply composed of the samples of the DTFT of the sequence at equally spaced frequency points, or equivalently, the samples of its z-transform at equally spaced points on the unit circle. The DFT provides information about the spectral contents of the signal at equally spaced discrete frequency points, and thus, can be used for spectral analysis of signals. Various techniques, commonly known as the fast Fourier transform (FFT) algorithms, have been advanced for the efficient computation of the DFT. An important tool in digital signal processing is the linear convolution of two finite-length signals, which often can be implemented very efficiently using the DFT.
A generalization of the discrete Fourier transform, introduced in this book, is the nonuniform discrete Fourier transform (NDFT), which can be used to obtain frequency domain information of a finite-length signal at arbitrarily chosen frequency points. Even though the NDFT concept has been alluded to by a number of authors in recent years and applied in particular to the design of one-dimensional (1-D) and two-dimensional (2-D) finite-impulse-response (FIR) digital filters, in this book, we provide a more formal introduction to the subject and discuss a number of interesting applications including the design of 1-D and 2-D FIR digital filters. We hope that including material on the NDFT and some of its signal processing applications in a single volume will generate more interest in this topic, and lead to many other applications in the field.
Introduction
The Nonuniform Discrete Fourier Transform
1-D FIR Filter Design Using the NDFT
2-D FIR Filter Design Using the NDFT
Antenna Pattern Synthesis with Prescribed Nulls
Dual-Tone Multi-Frequency Signal Decoding
Conclusions