The Nonuniform Discrete Fourier Transform and Its Applications in Signal Processing

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 sys tem 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 finite length 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 com putation 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.

1. Introduction.- 1.1 Overview.- 1.2 Discrete Fourier Transform.- 1.3 Chirp z-transform.- 1.4 Subband Discrete Fourier Transform.- 1.5 Computation of Nonuniformly Spaced Frequency Samples.- 1.6 Summary.- 2. The Nonuniform Discrete Fourier Transform.- 2.1 Basic Concepts.- 2.2 Properties of the NDFT.- 2.3 Computation of the NDFT.- 2.4 Subband NDFT.- 2.5 The 2-D NDFT.- 2.6 Summary.- 3. 1-D FIR Filter Design using the NDFT.- 3.1 Introduction.- 3.2 Existing Methods for Frequency Sampling Design.- 3.3 Proposed Nonuniform Frequency Sampling Design.- 3.4 Results.- 3.5 Summary.- 4. 2-D FIR Filter Design using the NDFT.- 4.1 Introduction.- 4.2 Existing Methods for 2-D Frequency Sampling.- 4.3 Proposed 2-D Nonuniform Frequency Sampling Design.- 4.4 Square Filter Design.- 4.5 Circularly Symmetric Filter Design.- 4.6 Diamond Filter Design.- 4.7 Ellipticaily-Shaped Lowpass Filter Design.- 4.8 Applications of 2-D Filters.- 4.9 Summary.- 5. Antenna Pattern Synthesis with Prescribed Nulls.- 5.1 Introduction.- 5.2 Existing Methods for Null Synthesis.- 5.3 Proposed Null Synthesis Method.- 5.4 Design Examples and Comparisons.- 5.5 Summary.- 6. Dual-Tone Multi-Frequency Signal Decoding.- 6.1 Introduction.- 6.2 Background.- 6.3 Proposed DTMF Decoding Algorithm Using the Subband NDFT.- 6.4 Results and Comparisons.- 6.5 Summary.- 7. Conclusions.- References.