Signal enhancement in the frequency domain

Mathematically this operation can be described by the same equation (eq. (40.13)) as derived for polynomial smoothing, namely  

Vq is called the cut-off frequency. H v) is referred in this context as a filter transfer function. 

Many other filter functions can be designed, e.g. an exponential or a trapezoidal function, or a band pass filter. As a rule exponential and trapezoidal filters perform better than cut-off filters, because an abrupt truncation of the Fourier coefficients may introduce artifacts, such as the annoying appearance of periodicities on the signal. The problem of choosing filter shapes is discussed in more detail by Lam and Isenhour [11] with references to a more thorough mathematical treatment of the subject. The expression for a band-pass filter is H v) = 1 for v j v else 

H v) = 0. This filter is particularly useful for removing periodic disturbances of the signal. 

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