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Notation

This appendix summarizes the notation used in this book.

Frequency and Time

  • $ \omega$ denotes continuous radian frequency (rad/sec)
  • $ f$ denotes continuous frequency in Hertz (Hz)
  • $ \omega = 2\pi f$
  • $ \omega_k$ denotes discrete frequency, $ \omega_k = 2\pi (k/N) f_s$
  • $ \omega, \omega_k \in {\bf R}$ (frequencies are always real)
  • $ T = $ sampling interval (sec)
  • $ f_s = $ sampling rate, $ f_s=\frac{1}{T}$
  • $ t_n = nT $ (discrete time)
  • $ n,k \in {\bf Z}$ (integers)
  • $ t,t_n \in {\bf R}$ (time is always real)

Signal Notation

  • $ x \in {\bf C}^N $ means $ x$ is a length $ N$ complex sequence
  • $ x = x(\cdot)$
  • $ X = \hbox{\sc DFT}(x) \in {\bf C}^N $
  • $ X(k) = \hbox{\sc DFT}_k(x) $
  • $ X(k) = \hbox{\sc DFT}_{N,k}(x)$ denotes the $ k$ th bin of a length $ N$ DFT
  • $ X(k) \in {\bf C}$
  • $ x(n) = \hbox{\sc IDFT}_n(X) $
  • $ x(n)\in{\bf R}$ or $ {\bf C}$
  • $ n,k \in {\bf Z}$ or $ n,k \in {\bf Z}_N$ (integers modulo $ N$ )
  • For $ x\in {\bf C}^\infty$ , $ X = \hbox{\sc DTFT}(x) \in {\bf C}_{2\pi}^{\infty}$
  • $ \overline{x} = $ conjugate of $ x$
  • $ \angle x = $ phase of $ x$

Fourier Transform Notation

The following notation will be used to state that $ X(\omega_k)$ is the Fourier Transform of $ x(t_n)$

$\displaystyle x \;\longleftrightarrow\;X \qquad [X=\hbox{\sc DFT}(x)]$ (A.1)

where $ \;\longleftrightarrow\;$ is read as ``corresponds to''. The notation $ XY$ or $ X\cdot Y$ denotes the vector containing $ (XY)_k=X(k)Y(k)$ , $ k=0,\ldots,N-1$ . This is denoted by `X .* Y' in Matlab, where X and Y are either column or row vectors.

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Selected Continuous Fourier Theorems
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Summary and Conclusions