scaleFilterSections - Scale cascaded transfer functions with scale values - MATLAB (original) (raw)

Scale cascaded transfer functions with scale values

Since R2023b

Syntax

Description

[Bg](#mw%5F6fb30b33-7740-412a-8ac6-d4c78493c702) = scaleFilterSections([B](#mw%5Fcb482135-57d0-45ae-9305-18af420209fc),[g](#mw%5F4f4d110d-3653-47d0-bcd3-bbd5c48f99ea)) scales the sections of the numerator filter coefficients B represented with Cascaded Transfer Functions (CTF) by applying the scale values specified in g.

example

Examples

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Design a 14th-order elliptic bandpass digital filter with a lower passband frequency of π/5 rad/sample and a higher passband frequency of 3π/5 rad/sample. Specify 10 dB of passband ripple and 40 dB of stopband attenuation. Represent the filter using cascaded transfer functions.

N = 14; [B,A] = ellip(N/2,10,40,[0.2 0.6],"ctf")

B = 4×5

0.4184    0.0000   -0.4184         0         0
0.4184   -0.3351    0.1684   -0.3351    0.4184
0.4184   -0.4070    0.3842   -0.4070    0.4184
0.4184   -0.4158    0.4107   -0.4158    0.4184

A = 4×5

1.0000   -0.7094    0.8572         0         0
1.0000   -1.1435    1.4172   -1.0733    0.9081
1.0000   -1.0196    1.0568   -1.0091    0.9862
1.0000   -1.0010    1.0029   -0.9998    0.9984

Define scale values as a vector with incremental gains (1, 2, ..., L) for each of the L sections, and append the overall system gain as 0.5L. This vector makes the numerator coefficients be scaled by 0.5, 1, ..., 0.5L.

numSections = size(B,1); g = [1:numSections 0.5^(numSections)];

Use scaleFilterSections to scale the filter numerator coefficients so that the gain is uniformly distributed across all sections.

Bg = scaleFilterSections(B,g)

Bg = 4×5

0.2092    0.0000   -0.2092         0         0
0.4184   -0.3351    0.1684   -0.3351    0.4184
0.6276   -0.6105    0.5764   -0.6105    0.6276
0.8367   -0.8317    0.8215   -0.8317    0.8367

Visualize the frequency response.

Input Arguments

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Unscaled CTF numerator filter coefficients, specified as a matrix.

• The number of rows in B equals L, where L is the number of filter sections in the cascade.
• The number of columns in B equals m + 1, where m is the section order.
• If you specify B as a vector,scaleFilterSections treats B as a matrix, determining the number of sections and the numerator order depending on vector size:
  • Row vector — The function treats B as a one-section transfer function with numerator order m, where m + 1 is the number of columns. The _i_th column of B corresponds to the _z_-(i - 1) numerator coefficient.
  • Column vector — The function treats B as an_L_-section transfer function with scalar numerators. Each row of B corresponds to a numerator in each corresponding section.

Data Types: single | double
Complex Number Support: Yes

Scale values, specified as a scalar or a vector with L + 1 elements, where L is the number of filter sections in the cascade.

• If g is a scalar, scaleFilterSections applies the value uniformly to all the cascade filter sections.
• If g is a vector, scaleFilterSections applies each of the first L scale values to the corresponding filter section and the last scale value uniformly to all the filter sections.

Data Types: single | double

Output Arguments

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Scaled CTF numerator filter coefficients, returned as a matrix of the same size asB.

See Algorithms for more information.

More About

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Partitioning an IIR digital filter into cascaded sections improves its numerical stability and reduces its susceptibility to coefficient quantization errors. The cascaded form of a transfer function H(z) in terms of the L transfer functions_H_1(z),_H_2(z), …,H L(z) is

Algorithms

The scaleFilterSections function scales the matrix B with a scalar or vector g and returns Bg as one of these:

• If g is a scalar:
  L = size(B,1);
  gL = (abs(g))^(1/L);
  Bg = B*gL;
  Bg(L,:) = sign(g)*Bg(L,:);
• If g is a vector with L+1 samples, given_L_ sections:
  L = size(B,1);
  gS = g(end);
  gL = (abs(gS))^(1/L);
  gl = g(1:end-1);
  Bg = B.*gl(:)*gL;
  Bg(L,:) = sign(gS)*Bg(L,:);

References

[1] Lyons, Richard G. Understanding Digital Signal Processing. Upper Saddle River, NJ: Prentice Hall, 2004.

Version History

Introduced in R2023b