0.1 Discrete-time signals  (Page 9/10)

The DFT values are samples of the DTFT of a finite length signal. The DTFT is the z-transform evaluated on the unit circle in the z plane.

and if $x\left(n\right)$ is of length $N$

It is important to be able to relate the time-domain signal $x\left(n\right)$ , its spectrum $X\left(\omega \right)$ , and its z-transform represented by the pole-zero locations on the z plane.

Relationships among fourier transforms

The DFT takes a periodic discrete-time signal into a periodic discrete-frequency representation.

The DTFT takes a discrete-time signal into a periodic continuous-frequency representation.

The FS takes a periodic continuous-time signal into a discrete-frequency representation.

The FT takes a continuous-time signal into a continuous-frequency representation.

The LT takes a continuous-time signal into a function of a continuous complex variable.

The ZT takes a discrete-time signal into a function of a continuous complex variable.

Wavelet-based signal analysis

There are wavelet systems and transforms analogous to the DFT, Fourier series, discrete-time Fourier transform, and the Fourier integral. Wewill start with the discrete wavelet transform (DWT) which is analogous to the Fourier series and probably should be called the wavelet series [link] . Wavelet analysis can be a form of time-frequency analysis which locates energy or events in time and frequency (or scale)simultaneously. It is somewhat similar to what is called a short-time Fourier transform or a Gabor transform or a windowed Fourier transform.

The history of wavelets and wavelet based signal processing is fairly recent. Its roots in signal expansion go back to earlygeophysical and image processing methods and in DSP to filter bank theory and subband coding. The current high interest probablystarted in the late 1980's with the work of Mallat, Daubechies, and others. Since then, the amount of research, publication, andapplication has exploded. Two excellent descriptions of the history of wavelet research and development are by Hubbard [link] and by Daubechies [link] and a projection into the future by Sweldens [link] and Burrus [link] .

The basic wavelet theory

The ideas and foundations of the basic dyadic, multiresolution wavelet systems are now pretty well developed, understood, andavailable [link] , [link] , [link] , [link] . The first basic requirement is that a set of expansion functions (usually a basis) aregenerated from a single “mother” function by translation and scaling. For the discrete wavelet expansion system, this is

where $j,k$ are integer indices for the series expansion of the form

The coefficients $c_{j,k}$ are called the discrete wavelet transform of the signal $f\left(t\right)$ . This use of translation andscale to create an expansion system is the foundation of all so-called first generation wavelets [link] .

The system is somewhat similar to the Fourier series described in Equation 51 from Least Squared Error Designed of FIR Filters with frequencies being related by powers of two rather than an integer multiple and the translation by $k$ giving only the two results of cosine and sine for the Fourier series.