designMultirateFIR

Multirate FIR filter design

Syntax

B = designMultirateFIR(L,M)

B = designMultirateFIR(L,M,P)

B = designMultirateFIR(L,M,TW)

B = designMultirateFIR(L,M,P,Astop)

B = designMultirateFIR(L,M,TW,Astop)

B = designMultirateFIR(___,'SystemObject',flag)

Description

B = designMultirateFIR(L,M) designs a multirate FIR filter with interpolation factor L and decimation factor M. The output B is the vector of designed FIR coefficients. To design a pure interpolator, set M to 1. To design a pure decimator, set L to 1.

B = designMultirateFIR(L,M,P) designs a multirate FIR filter with half-polyphase length P. By default, the half-polyphase length is 12.

B = designMultirateFIR(L,M,TW) designs a multirate FIR filter with interpolation factor L, decimation factor M, and normalized transition width TW.

B = designMultirateFIR(L,M,P,Astop) designs a multirate FIR filter with stopband attenuation Astop. By default, the stopband attenuation is 80 dB.

B = designMultirateFIR(L,M,TW,Astop) designs a multirate FIR filter with interpolation factor L, decimation factor M, normalized transition width TW, and stopband attenuation Astop, specified in dB.

B = designMultirateFIR(___,'SystemObject',flag) returns a vector of filter coefficients B if the flag is set to false, or a multirate filter System object™ if the flag is set to true.

Examples

Design an FIR Interpolator

To design an FIR Interpolator using the designMultirateFIR function, you must specify the interpolation factor of interest (usually greater than 1) and a decimation factor equal to 1. You can use the default half-polyphase length of 12 and the default stopband attenuation of 80 dB. Alternately, you can also specify the half-polyphase length and stopband attenuation values.

Design an FIR interpolator with interpolation factor set to 5. Use the default half-polyphase length of 12 and the default stopband attenuation of 80 dB.

b = designMultirateFIR(5,1);
fvtool(b,'impulse')

Design an FIR Decimator

To design an FIR Decimator using the designMultirateFIR function, you must specify the decimation factor of interest (usually greater than 1) and an interpolation factor equal to 1. You can use the default half-polyphase length of 12 and the default stopband attenuation of 80 dB. Alternately, you can also specify the half-polyphase length and stopband attenuation values. Design an FIR decimator with decimation factor set to 3, and half-polyphase length set to 14. Use the default stopband attenuation of 80 dB.

b = designMultirateFIR(1,3,14);
fvtool(b,'impulse');

Design an FIR Rate Converter

To design an FIR Rate Converter using the designMultirateFIR function, you must specify the interpolation and decimation factors of interest (usually greater than 1). In addition, you can specify either the half-polyphase length and stopband attenuation values, or the normalized transition width and stopband attenuation values.

Design an FIR rate converter with interpolation factor set to 3, decimation factor set to 4, half-polyphase length set to 14, and stopband attenuation set to 90 dB.

b = designMultirateFIR(3,4,14,90);
impz(b,1)

Design an FIR rate converter with interpolation factor set to 3, decimation factor set to 4, normalized transition width set to 0.2, and stopband attenuation set to 90 dB.

bTW = designMultirateFIR(3,4,0.2,90);
impz(bTW,1)

Design a dsp.FIRInterpolator System object

Set the 'SystemObject' flag to true in the designMultirateFIR function to design a multirate filter object. The design parameters specified in the function determine the type of System object the function designs.

In this example, the function designs a polyphase FIR interpolator System object™. For more details, see dsp.FIRInterpolator.

Create a dsp.FIRInterpolator object with the interpolation factor equal to 5, transition width equal to 0.01, and stopband attenuation equal to 60 dB. Set the 'SystemObject' flag to true to design a multirate filter object.

firInterp = designMultirateFIR(5,1,0.01,60,'SystemObject',true);
fvtool(firInterp)

Compute the cost of implementing the filter.

cost(firInterp)

ans = struct with fields:
                  NumCoefficients: 582
                        NumStates: 145
    MultiplicationsPerInputSample: 582
          AdditionsPerInputSample: 578

Measure the frequency response characteristics of the filter object.

measure(firInterp)

ans = 
Sample Rate      : N/A (normalized frequency)
Passband Edge    : 0.195                     
3-dB Point       : 0.19884                   
6-dB Point       : 0.2                       
Stopband Edge    : 0.205                     
Passband Ripple  : 0.016474 dB               
Stopband Atten.  : 60.1827 dB                
Transition Width : 0.01                      
 

Input Arguments

L — Interpolation factor

positive scalar integer

Interpolation factor, specified as a positive scalar integer. To design a decimator only, set L to 1.

Example: 2

Data Types: single | double | int8 | int16 | int32 | int64 | uint8 | uint16 | uint32 | uint64

M — Decimation factor

positive scalar integer

Decimation factor, specified as a positive scalar integer. To design an interpolator only, set M to 1.

Example: 2

Data Types: single | double | int8 | int16 | int32 | int64 | uint8 | uint16 | uint32 | uint64

P — Half-polyphase length

12 (default) | positive scalar integer

Half-polyphase length, specified as a positive scalar integer that is greater than or equal to 1.

Example: 12

Example: 20

Data Types: single | double | int8 | int16 | int32 | int64 | uint8 | uint16 | uint32 | uint64

TW — Normalized transition width

real scalar in the range (0 1)

Normalized transition width of the multirate FIR filter, specified as a real scalar in the range (0 1).

Data Types: single | double | int8 | int16 | int32 | int64 | uint8 | uint16 | uint32 | uint64

Astop — Stopband attenuation

90 (default) | nonnegative scalar

Stopband attenuation in dB, specified as a nonnegative real scalar greater than or equal to 0.

Example: 0.0

Example: 80.5

Data Types: single | double | int8 | int16 | int32 | int64 | uint8 | uint16 | uint32 | uint64

flag — System object flag

false (default) | true

System object flag set to:

• false –– Function returns a vector of filter coefficients.

• true –– Function returns one of the following multirate filter System objects:

  • dsp.FIRInterpolator –– When L > 1 and M = 1.

  • dsp.FIRDecimator –– When L = 1 and M > 1.

  • dsp.FIRRateConverter –– When L > 1 and M > 1.

Data Types: logical

Output Arguments

B — Coefficients

real-valued vector

Multirate FIR filter coefficients, returned as a real-valued N-length vector.

If both L and M are equal to 1, then N equals 1.

If L > 1 or M > 1, then N is given by $$N=\{\begin{array}{c}2PR+1\\ 2PR\end{array}\begin{array}{l}M>L>1\text{and}\mathrm{mod}(PL,M)\ne 0\hfill \\ \text{otherwise}\hfill \end{array}$$, where P is the half-polyphase length and R is defined by the following equations:

• If L > 1, R = L.

• If L = 1, R = M.

For more details, see the Algorithms section.

When the 'SystemObject' flag is set to true, the function returns one of the following multirate filter System objects:

• dsp.FIRInterpolator –– When L > 1 and M = 1.

• dsp.FIRDecimator –– When L = 1 and M > 1.

• dsp.FIRRateConverter –– When L > 1 and M > 1.

Data Types: double

Algorithms

designMultirateFIR designs an R^th band Nyquist FIR filter using a Kaiser window vector to window the truncated impulse response of the FIR filter.

The filter length N is defined as one of the following:

$$N=\{\begin{array}{c}2PR+1\\ 2PR\end{array}\begin{array}{l}M>L>1\text{and}\mathrm{mod}(PL,M)\ne 0\hfill \\ \text{otherwise}\hfill \end{array}$$

P is the half-polyphase length and R is defined as explained in B.

The truncated impulse response d(n) is delayed by N/2 samples to make it causal. The truncated and delayed impulse response is given by:

$$d(n-N/2)=\frac{\sin (w_c(n-N/2))}{\pi (n-N/2)},n=0,\cdots ,\frac{N}{2},\cdots ,N$$

where $$w_c=\pi /R$$.

For every R^th band, the impulse response of the Nyquist filters is exactly zero. Because of this property, when Nyquist filters are used for pure interpolation, the input samples remain unaltered after interpolating.

A Kaiser window is used because of its near-optimum performance while providing a robust way of designing a Nyquist filter. The window depends on two parameters: length N + 1 and shape parameter β.

The Kaiser window is defined by:

$$w(n)=\frac{I_0\left(\beta \sqrt{1-{\left(\frac{n-N/2}{N/2}\right)}^2}\right)}{I_0(\beta )},0\le n\le N,$$

where I₀ is the zeroth-order modified Bessel function of the first kind.

The shape parameter β is calculated from:

$$\beta =\{\begin{array}{ll}0.1102(A_{stop}-8.7)\hfill & \text{if }{A}_{stop}\ge 50\hfill \\ 0.5842{(A_{stop}-21)}^{0.4}+0.07886(A_{stop}-21)\hfill & \text{if }21<A_{stop}<50\hfill \\ 0\hfill & \text{if }{A}_{stop}\le 21,\hfill \end{array}$$

where A_stop is the stopband attenuation in dB.

The windowed impulse response is given by

$$h(n)=w(n)d(n-N/2)=w(n)\frac{\sin (w_c(n-N/2))}{\pi (n-N/2)},n=0,\cdots ,\frac{N}{2},\cdots ,N$$

h(n) for n = 0,..,N/2,...N are the coefficients of the multirate filter. These coefficients are defined by the interpolation factor, L, and decimation factor, M.

References

[1] Orfanidis, Sophocles J. Introduction to Signal Processing. Upper Saddle River, NJ: Prentice-Hall, 1996.

Extended Capabilities

C/C++ Code Generation

Generate C and C++ code using MATLAB® Coder™.

Usage notes and limitations:

• The function must return a vector of coefficients. Setting the 'SystemObject' flag to true is not supported for code generation.

• The inputs to the function must be constants when generating code for transition width-based filter design. In this design, you cannot specify the polyphase length (and therefore the filter length) and the function determines this value iteratively.

  While generating code from the designMultirateFIR function, if you specify the inputs to the function as:

  • Constants –– The third argument (if specified) is either treated as the transition width or the half polyphase length. If the value of the third argument is less than 1, then it is treated as the transition width. Otherwise, the third argument is treated as the half-polyphase length. This behavior is consistent with the behavior of the function.

    % Transition width equals 0.1
    codegen designMultirateFIR -args {coder.Constant(5),coder.Constant(4),coder.Constant(0.1)}
    designMultirateFIR_mex(5,4,.1)

    % Half-polyphase length equals 10
    codegen designMultirateFIR -args {coder.Constant(5),coder.Constant(4),coder.Constant(10)}
    designMultirateFIR_mex(5,4,10)

  • Nonconstants –– The third argument (if specified) is always treated as the half-polyphase length, irrespective of its value.

    For example, specify the third argument as 0.1 while generating the code. Even though the function treats a value less than 1 as the transition width, the generated code treats this value as the half-polyphase length.

    codegen designMultirateFIR -args {5,4,0.1}

    While running the generated code, specify the third argument as a positive integer since the half-polyphase length input must be a positive integer.

    designMultirateFIR_mex(5,4,1)

    However, if you specify the third argument as a scalar less than 1, the function displays the following error message.

    designMultirateFIR_mex(5,4,0.1)

    Expected P to be a scalar with value >= 1.

Version History

Introduced in R2016a