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Nyquist filters have the following magnitude response specifications:

Filter Specification	Value Range
Passband edge frequency	[0, fs/2M−ε]
Stopband edge frequency	[fs/2M+ε, fs/2]

In this table, M denotes the sampling frequency conversion factor and f_s denotes the sampling frequency of a Nyquist filter. For an interpolation Nyquist filter, f_s equals L times the sampling frequency of the input signal, where L denotes the interpolation factor. For a decimation Nyquist filter, f_s equals the sampling frequency of the input signal. You can specify ε indirectly by using the roll off , which is defined as the following equation:

The following figure illustrates the magnitude response of a Nyquist filter.

The following sections describe how to use the LabVIEW Digital Filter Design Toolkit to design Nyquist filters, including raised cosine filters and halfband filters.

Designing Nyquist Filters

Nyquist filters, also called M^th band filters, are a special type of multirate finite impulse response (FIR) filter. Nyquist filter coefficients have periodic zero values every M^th sample, except for the middle coefficient. The following figure shows the coefficients of a Nyquist filter with a sampling frequency conversion factor of 4. The 26^th coefficient is the middle coefficient and does not have a zero value.

The impulse response of a Nyquist filter h(n) satisfies the following equation:

where c and k are constants.

The z-transform of a Nyquist filter H(z) satisfies the following equation:

where W = e^−j2π/M and c = 1/M. The frequency response of H(zW^k) is the shifted version of the frequency response of H(z), so the frequency responses of all M uniformly shifted versions of H(z) add up to a constant.

Use the DFD Nyquist Design VI to design Nyquist filters. You can use Nyquist filters to remove images in interpolation. Nyquist filters modify the interpolated zeroes, but they do not change the original samples.

Designing Raised Cosine Filters

Raised cosine filters are a special case of Nyquist filters. As with other Nyquist filters, the coefficients of the raised cosine filter have periodic zero values every M^th sample except for the middle coefficient. The ideal frequency response of a raised cosine filter consists of unity gain at low frequencies, a raised cosine shape in the middle, and zero gain at high frequencies. The following equation describes the magnitude response of a raised cosine filter.

where f_c is the cutoff frequency and is the roll off, which satisfies 0 ≤  ≤ 1.

In digital communication systems, if you want to split the overall raised cosine filtering evenly between the transmitter filter and receiver filter, use root-raised cosine filters. The following equation describes the magnitude response of a root-raised cosine filter.

Use the DFD Raised Cosine Design VI to design raised cosine filters and root-raised cosine filters.

Designing Halfband Filters

Halfband filters are M^th band filters when M = 2. The impulse response of a halfband filter h(n) satisfies the following equation:

where c and k are constants. c usually equals 0.5.

The magnitude response of a halfband filter is symmetric with respect to the frequency f_s/4. The value of the magnitude response at f_s/4 is fixed to 0.5. The filter has a linear phase property. Nearly half of the filter coefficients in a halfband filter are zeroes, which greatly reduces the computations required for filtering. Use the DFD Halfband Design VI to design halfband filters.

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