freqz

CWT filter bank frequency responses

Syntax

[psidft,f] = freqz(fb)

[___] = freqz(___,Name=Value)

freqz(___)

Description

[psidft,f] = freqz(fb) returns the frequency responses for the wavelet filters, psidft, and the frequency vector, f, for the continuous wavelet transform (CWT) filter bank, fb. Frequencies are in cycles/sample or Hz. If you specify a sampling period, the frequencies are in cycles/unit time where the time unit is the unit of the duration sampling period.

The frequency responses, psidft, are one-sided frequency responses for the positive frequencies. For the analytic wavelets supported by cwtfilterbank, the frequency responses are real-valued and are equivalent to the magnitude frequency response.

[___] = freqz(___,Name=Value) specifies one or more additional name-value arguments. For example, psidft = freqz(fb,FrequencyRange="twosided") returns the full two-sided frequency responses.

freqz(___) with no output arguments plots the magnitude frequency responses for the CWT filter bank, fb.

Examples

Invert CWT Using Approximate Synthesis Filters

Load the Kobe earthquake data. Create a CWT filter bank with period boundary handling that you can apply to the data.

load kobe
fb = cwtfilterbank(SignalLength=numel(kobe),Boundary="periodic");

Obtain the two-sided wavelet and scaling filter responses.

[psidft,f] = freqz(fb,IncludeLowpass=true,FrequencyRange="twosided");

Obtain the CWT of the data. Also obtain the scaling coefficients.

[cfs,~,~,scalcfs] = wt(fb,kobe);

Invert the transform using the filter bank and the scaling coefficients.

xrec = icwt(cfs,ScalingCoefficients=scalcfs,AnalysisFilterBank=psidft);
plot([kobe(:) xrec(:)])
axis tight

Obtain the maximum reconstruction error.

norm(kobe(:)-xrec(:),'Inf')

ans = 
2.9104e-11

Frequency Responses of Continuous Wavelet Transform Filter Bank

Create a CWT filter bank. Set the voices per octave to 14, the sampling frequency to 1000 Hz, and frequency limits to range from 200 Hz to 300 Hz.

fb = cwtfilterbank(VoicesPerOctave=14,...
    SamplingFrequency=1000,FrequencyLimits=[200 300]);

Plot the frequency responses.

freqz(fb)

Boundary Handling and Frequency Range

This example shows how boundary handling and signal length affect the range of frequency responses freqz returns.

Reflection / Even Length

Create a CWT filter bank suitable for an even-length signal. Use the default Boundary setting reflection.

sLen = 256;
fb = cwtfilterbank(SignalLength=sLen);

Obtain the one-sided frequency responses of the filter bank. Also obtain the frequency vector.

[psidft,f] = freqz(fb,FrequencyRange="onesided");

Confirm the range of frequencies includes the Nyquist.

f(end)

ans = 
0.5000

Plot the frequency response of the filter with the highest center frequency.

plot(f,psidft(1,:))
xlabel("Frequency (samples/cycle)")
ylabel("Magnitude")
title("Reflection / Even / One-sided")

Obtain and plot the two-sided frequency responses. Confirm the frequency range does not include the Nyquist.

[psidft,f] = freqz(fb,FrequencyRange="twosided");
f(end)

ans = 
0.9980

plot(f,psidft(1,:))
xlabel("Frequency (samples/cycle)")
ylabel("Magnitude")
title("Reflection / Even / Two-sided")

Reflection / Odd Length

Create a CWT filter bank suitable for an odd-length signal. Use the default Boundary setting reflection.

sLen = 255;
fb = cwtfilterbank(SignalLength=sLen);

Obtain the one-sided frequency responses of the filter bank. Confirm the range of frequencies does not include the Nyquist.

[~,f] = freqz(fb,FrequencyRange="onesided");
f(end)

ans = 
0.4990

Periodic / Even Length

Create a CWT filter bank with periodic boundary handling suitable for an even-length signal

sLen = 256;
fb = cwtfilterbank(SignalLength=sLen,Boundary="periodic");

Obtain the one-sided frequency responses of the filter bank. Confirm the range of frequencies does include the Nyquist.

[~,f] = freqz(fb,FrequencyRange="onesided");
f(end)

ans = 
0.5000

Periodic / Odd Length

Create a CWT filter bank with periodic boundary handling suitable for an odd-length signal

sLen = 255;
fb = cwtfilterbank(SignalLength=sLen,Boundary="periodic");

Obtain the one-sided frequency responses of the filter bank. Confirm the range of frequencies does not include the Nyquist.

[~,f] = freqz(fb,FrequencyRange="onesided");
f(end)

ans = 
0.4980

Input Arguments

fb — Continuous wavelet transform filter bank

cwtfilterbank object

Continuous wavelet transform (CWT) filter bank, specified as a cwtfilterbank object.

Name-Value Arguments

Specify optional pairs of arguments as Name1=Value1,...,NameN=ValueN, where Name is the argument name and Value is the corresponding value. Name-value arguments must appear after other arguments, but the order of the pairs does not matter.

Example: psidft = freqz(fb,IncludeLowpass=true) appends the lowpass, or scaling filter, frequency response as the final row of psidft.

IncludeLowpass — Append lowpass filter frequency response

false or 0 (default) | true or 1

Option to append lowpass, or scaling filter, frequency response as the final row of psidft, specified as one of these:

• 1 (true) — Include the frequency response symmetrically

• 0 (false) — Do not include the frequency response

For the analytic wavelets supported by cwtfilterbank, the scaling filter frequency response is real-valued and is equivalent to the magnitude frequency response.

Data Types: logical

FrequencyRange — Frequency range for filter responses

"onesided" (default) | "twosided"

Frequency range for the wavelet and scaling function frequency responses, specified as one of "onesided", or "twosided". The frequency ranges corresponding to each option are

• "onesided" — returns the frequency responses from [0,½] when the length of the padded filters is even and [0,½) when the length of the padded filters is odd. Padding is added when the Boundary property of the filter bank is "reflection".

  If a sampling frequency Fs is specified in the filter bank, the intervals become [0,Fs/2] and [0,Fs/2) respectively.

• "twosided" — returns the full two-sided frequency responses over the range [0,1). If a sampling frequency Fs is specified in the filter bank, the interval becomes [0,Fs).

Note

To use the wavelet and scaling filters in the inverse CWT, set Boundary in the filter bank to "periodic", and use IncludeLowpass=true and FrequencyRange="twosided" in freqz.

Output Arguments

psidft — Frequency responses

real-valued 2-D matrix

Frequency responses of a CWT filter bank, returned as a real-valued matrix. Each column of psidft is the response at the frequency in the corresponding element of f.

By default, frequency responses, psidft, are one-sided frequency responses for the positive frequencies. For the analytic wavelets supported by cwtfilterbank, the frequency responses are real-valued and are equivalent to the magnitude frequency response.

Data Types: double

f — Frequencies

real-valued vector

Frequencies, in cycles/sample or hertz, returned as a real-valued vector.

If you specify a sampling period, the frequencies are in cycles/unit time, where the time unit is the unit of the duration SamplingPeriod.

Data Types: double

Extended Capabilities

C/C++ Code Generation

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

Usage notes and limitations:

• Plotting is not supported.

Version History

Introduced in R2018a