0.6 Digital filtering and the dft  (Page 3/13)

Many other aspects of continuous-time signals and systems have analogs in discrete time. Following are some that will be usefulin later chapters:

• Symmetry—If the time signal $w\left[k\right]$ is real, then $W^{*}\left[n\right]=W[N-n]$ . This is analogous to [link] .
• Parseval's theorem holds in discrete time— ${\sum }_k{w}^2\left[k\right]=\frac{1}{N}{\sum }_n{\left|W\left[n\right]\right|}^2$ . This is analogous to [link] .
• The frequency response $H\left[n\right]$ of a LTI system is the DFT of the impulse response $h\left[k\right]$ . This is analogous to the continuous-time result that the frequency response $H\left(f\right)$ is the Fourier transform of the impulse response $h\left(t\right)$ .
• Time delay property in discrete time— $w[k-l]\iff W\left[n\right]e^{-j(2\pi /N)l}$ . This is analogous to [link] .
• Modulation property—This frequency shifting property is analogous to [link] .
• If $w\left[k\right]=sin\left(\frac{2\pi fk}{T}\right)$ is a periodic sine wave, then the spectrum is a sum of two delta impulses.This is analogous to the result in Example [link] .
• Convolution To be precise, this should be circular convolution . However, for the purposes of designing a workable receiver,this distinction is not essential. The interested reader can explore the relationship ofdiscrete-time convolution in the time and frequency domains in a concrete way using waystofilt.m . in (discrete) time is the same as multiplication in (discrete) frequency.This is analogous to [link] .
• Multiplication in (discrete) time is the same as convolution in (discrete) frequency.This is analogous to [link] .
• The transfer function of a LTI system is the ratio of the DFT of the output and the DFT of the input. This is analogous to [link] .

Understanding the dft

Define a vector $\mathbf{W}$ containing all $N$ frequency values and a vector $\mathbf{w}$ containing all $N$ time values

and let $M^{-1}$ be a matrix with columns of complex exponentials

Then the IDFT [link] can be rewritten as a matrix multiplication

and the DFT is

Since the inverse of an orthonormal matrix is equal to its own complex conjugate transpose, $M$ in [link] is the same as $M^{-1}$ in [link] with the signs on all the exponents flipped.

The matrix $M^{-1}$ is highly structured. Let $C_n$ be the $n^{th}$ column of $M^{-1}$ . Multiplying both sides by $N$ , [link] can be rewritten as

This form displays the time vector $\mathbf{w}$ as a linear combination Those familiar with advanced linear algebra will recognize that $M^{-1}$ can be thought of as a change of basis that reexpresses $\mathbf{w}$ in a basis defined by the columns of $M^{-1}$ . of the columns $C_n$ . What are these columns? They are vectors of discrete (complex valued) sinusoids, each at a differentfrequency. Accordingly, the DFT reexpresses the time vector as a linear combination of these sinusoids. The complexscaling factors $W\left[n\right]$ define how much of each sinusoid is present in the original signal $w\left[k\right]$ .