By M. H Greenblatt

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**Additional info for Mathematical entertainments: A collection of illuminating puzzles**

**Example text**

This property will allow us to turn the difference equation of a digital filter into an algebraic equation and then determine the mathematical description of a digital filter, called its transfer function. Also, if we have determined the mathematical description, or transfer function, of a digital filter, we can use the shifting property to write the difference equation of the filter that is actually coded. 1 gives the input signal into an ADC in column 1, the corresponding sampled signal in column 2, and the z-transform of that signal in column 3 for several signals using the definition of the z-transform.

The filter is to reduce all frequencies below 10 Hz and above 500 Hz by more than 60 dB, while not reducing the frequencies between 50 Hz and 100 Hz by more than 2 dB. The filter should also not increase any frequency in the passband by more than 1 dB. Solution: First the sample time T is used to determine the folding frequency. wf = π = 6283 rad/s T All the other frequencies are multiplied by 2π to convert to rad/s. 6. 6. 4, it can be seen that the graphical specification for a highpass digital filter that is plotted out to the sampling frequency would have the same form as a bandpass filter graphical specification plotted only out to half the sampling frequency.

This is 1,000π rad/s. The filter is required to reduce input frequencies below 200 rad/s by more than 20 dB, but not reduce any input frequencies above 500 rad/s by more than 1 dB and not increase any signals above that. 4. Any filter designed using the methods shown in Chapters 7, 10, and 11 that has a gain in the clear area meets the customer’s specification for the filter. 3. 4. 4 The Bandpass Digital Filter Specification The digital bandpass filter specification is a little more complex than the previous two graphical specifications simply because the allowed filter gain region is more complex.