**Lowpass FIR Filter -- Window Method**

This example shows how to design and implement an FIR filter using two command line functions, `fir1`

and `designfilt`

, and the interactive tool `fdatool`

.

Create a signal to use in the examples. The signal is a 100 Hz sine wave in additive white Gaussian noise. Set the random number generator to the default state for reproducible results.

```
rng default
Fs = 1000;
t = linspace(0,1,Fs);
x = cos(2*pi*100*t)+0.5*randn(size(t));
```

The filter design is an FIR lowpass filter with order equal to 20 and a cutoff frequency of 150 Hz. Use a Kaiser window with length one sample greater than the filter order and
. See `kaiser`

for details on the Kaiser window.

Use `fir1`

to design the filter. `fir1`

requires normalized frequencies in the interval [0,1], where 1 corresponds to
rad/sample. To use `fir1`

, you must convert all frequency specifications to normalized frequencies.

Design the filter and view the filter's magnitude response.

fc = 150; Wn = (2/Fs)*fc; b = fir1(20,Wn,'low',kaiser(21,3)); fvtool(b,1,'Fs',Fs)

Apply the filter to the signal and plot the result for the first ten periods of the 100 Hz sinusoid.

y = filter(b,1,x); plot(t(1:100),x(1:100)) hold on plot(t(1:100),y(1:100)) xlabel('Time (s)') ylabel('Amplitude') legend('Original Signal','Filtered Data')

Design the same filter using `designfilt`

. Set the filter response to `'lowpassfir'`

and input the specifications as `Name,Value`

pairs. With `designfilt`

, you can specify your filter design in Hz.

Fs = 1000; Hd = designfilt('lowpassfir','FilterOrder',20,'CutoffFrequency',150, ... 'DesignMethod','window','Window',{@kaiser,3},'SampleRate',Fs);

Filter the data and plot the result.

y1 = filter(Hd,x); figure plot(t(1:100),x(1:100)) hold on plot(t(1:100),y1(1:100)) xlabel('Time (s)') ylabel('Amplitude') legend('Original Signal','Filtered Data')

**Lowpass FIR Filter with FDATool**

This example shows how to design and implement a lowpass FIR filter using the window method with the interactive tool `fdatool`

.

Start FDATool by entering

`fdatool`

at the command line.Set the

**Response Type**to**Lowpass**.Set the

**Design Method**to**FIR**and select the**Window**method.Under

**Filter Order**, select**Specify order**. Set the order to 20.Under

**Frequency Specifications**, set**Units**to**Hz**,**Fs**to 1000, and**Fc**to 150.

Click

**Design Filter**.Select

**File**>**Export...**to export your FIR filter to the MATLAB® workspace as coefficients or a filter object. In this example, export the filter as an object. Specify the variable name as`Hd1`

.

Click

**Export**.Filter the input signal in the command window with the exported filter object. Plot the result for the first ten periods of the 100 Hz sinusoid.

y2 = filter(Hd,x); figure plot(t(1:100),x(1:100)) hold on plot(t(1:100),y2(1:100)) xlabel('Time (s)') ylabel('Amplitude') legend('Original Signal','Filtered Data')

Select

**File**>**Generate MATLAB Code**to generate a MATLAB function to create a filter object using your specifications.

You can also use the interactive tool `filterbuilder`

to design your filter.

**Bandpass Filters -- Minimum-Order FIR and IIR Systems**

This example shows how to design a bandpass filter and filter data with minimum-order FIR equiripple and IIR Butterworth filters. You can model many real-world signals as a superposition of oscillating components, a low-frequency trend, and additive noise. For example, economic data often contain oscillations, which represent cycles superimposed on a slowly varying upward or downward trend. In addition, there is an additive noise component, which is a combination of measurement error and the inherent random fluctuations in the process.

In these examples, assume you sample some process every day for one year. Assume the process has oscillations on approximately one-week and one-month scales. In addition, there is a low-frequency upward trend in the data and additive white Gaussian noise.

Create the signal as a superposition of two sine waves with frequencies of 1/7 and 1/30 cycles/day. Add a low-frequency increasing trend term and white Gaussian noise. Set the random number generator to the default state for reproducible results. The data is sampled at 1 sample/day. Plot the resulting signal and the power spectral density (PSD) estimate.

rng default Fs = 1; n = 1:365; x = cos(2*pi*(1/7)*n)+cos(2*pi*(1/30)*n-pi/4); trend = 3*sin(2*pi*(1/1480)*n); y = x+trend+0.5*randn(size(n)); [pxx,f] = periodogram(y,[],length(y),Fs); subplot(2,1,1) plot(n,y) xlim([1 365]) xlabel('Days') grid subplot(2,1,2) plot(f,10*log10(pxx)) xlabel('Cycles/day') ylabel('dB') grid

The low-frequency trend appears in the power spectral density estimate as increased low-frequency power. The low-frequency power appears approximately 10 dB above the oscillation at 1/30 cycles/day. Use this information in the specifications for the filter stopbands.

Design minimum-order FIR equiripple and IIR Butterworth filters with the following specifications: passband from [1/40,1/4] cycles/day and stopbands from [0,1/60] and [1/4,1/2] cycles/day. Set both stopband attenuations to 10 dB and the passband ripple tolerance to 1 dB.

Hd1 = designfilt('bandpassfir', ... 'StopbandFrequency1',1/60,'PassbandFrequency1',1/40, ... 'PassbandFrequency2',1/4 ,'StopbandFrequency2',1/2 , ... 'StopbandAttenuation1',10,'PassbandRipple',1, ... 'StopbandAttenuation2',10,'DesignMethod','equiripple','SampleRate',Fs); Hd2 = designfilt('bandpassiir', ... 'StopbandFrequency1',1/60,'PassbandFrequency1',1/40, ... 'PassbandFrequency2',1/4 ,'StopbandFrequency2',1/2 , ... 'StopbandAttenuation1',10,'PassbandRipple',1, ... 'StopbandAttenuation2',10,'DesignMethod','butter','SampleRate',Fs);

Compare the order of the FIR and IIR filters and the unwrapped phase responses.

fprintf('The order of the FIR filter is %d\n',filtord(Hd1)) fprintf('The order of the IIR filter is %d\n',filtord(Hd2)) [phifir,w] = phasez(Hd1,[],1); [phiiir,w] = phasez(Hd2,[],1); figure plot(w,unwrap(phifir)) hold on plot(w,unwrap(phiiir)) xlabel('Cycles/Day') ylabel('Radians') legend('FIR Equiripple Filter','IIR Butterworth Filter') grid

The order of the FIR filter is 78 The order of the IIR filter is 8

The IIR filter has a much lower order that the FIR filter. However, the FIR filter has a linear phase response over the passband, while the IIR filter does not. The FIR filter delays all frequencies in the filter passband equally, while the IIR filter does not.

Additionally, the rate of change of the phase per unit of frequency is greater in the FIR filter than in the IIR filter.

Design a lowpass FIR equiripple filter for comparison. The lowpass filter specifications are: passband [0,1/4] cycles/day, stopband attenuation equal to 10 dB, and the passband ripple tolerance set to 1 dB.

Hdlow = designfilt('lowpassfir', ... 'PassbandFrequency',1/4,'StopbandFrequency',1/2, ... 'PassbandRipple',1,'StopbandAttenuation',10, ... 'DesignMethod','equiripple','SampleRate',1);

Filter the data with the bandpass and lowpass filters.

yfir = filter(Hd1,y); yiir = filter(Hd2,y); ylow = filter(Hdlow,y);

Plot the PSD estimate of the bandpass IIR filter output. You can replace `yiir`

with `yfir`

in the following code to view the PSD estimate of the FIR bandpass filter output.

[pxx,f] = periodogram(yiir,[],length(yiir),Fs); figure plot(f,10*log10(pxx)) xlabel('Cycles/day') ylabel('dB') grid

The PSD estimate shows the bandpass filter attenuates the low-frequency trend and high-frequency noise.

Plot the first 120 days of FIR and IIR filter output.

figure plot(n(1:120),yfir(1:120)) hold on plot(n(1:120),yiir(1:120)) axis([1 120 -2.8 2.8]) xlabel('Days') legend('FIR bandpass filter output','IIR bandpass filter output', ... 'Location','SouthEast')

The increased phase delay in the FIR filter is evident in the filter output.

Plot the lowpass FIR filter output superimposed on the superposition of the 7-day and 30-day cycles for comparison.

figure plot(n,x) hold on plot(n,ylow) xlim([1 365]) xlabel('Days') legend('7-day and 30-day cycles','FIR lowpass filter output', ... 'Location','NorthWest')

You can see in the preceding plot that the low-frequency trend is evident in the lowpass filter output. While the lowpass filter preserves the 7-day and 30-day cycles, the bandpass filters perform better in this example because the bandpass filters also remove the low-frequency trend.

**Zero-Phase Filtering**

This example shows how to perform zero-phase filtering.

Repeat the signal generation and lowpass filter design with `fir1`

and `designfilt`

. You do not have to execute the following code if you already have these variables in your workspace.

rng default Fs = 1000; t = linspace(0,1,Fs); x = cos(2*pi*100*t)+0.5*randn(size(t)); % Using fir1 fc = 150; Wn = (2/Fs)*fc; b = fir1(20,Wn,'low',kaiser(21,3)); % Using designfilt Hd = designfilt('lowpassfir','FilterOrder',20,'CutoffFrequency',150, ... 'DesignMethod','window','Window',{@kaiser,3},'SampleRate',Fs);

Filter the data using `filter`

. Plot the first 100 points of the filter output along with a superimposed sinusoid with the same amplitude and initial phase as the input signal.

yout = filter(Hd,x); xin = cos(2*pi*100*t); figure plot(t(1:100),xin(1:100)) hold on plot(t(1:100),yout(1:100)) xlabel('Time (s)') ylabel('Amplitude') legend('Input Sine Wave','Filtered Data', ... 'Location','NorthEast') grid

Looking at the initial 0.01 seconds of the filtered data, you see that the output is delayed with respect to the input. The delay appears to be approximately 0.01 seconds, which is almost 1/2 the length of the FIR filter in samples .

This delay is due to the filter's phase response. The FIR filter in these examples is a type I linear-phase filter. The group delay of the filter is 10 samples.

Plot the group delay using `fvtool`

.

fvtool(Hd,'Analysis','grpdelay');

In many applications, phase distortion is acceptable. This is particularly true when phase response is linear. In other applications, it is desirable to have a filter with a zero-phase response. A zero-phase response is not technically possibly in a noncausal filter. However, you can implement zero-phase filtering using a causal filter with `filtfilt`

.

Filter the input signal using `filtfilt`

. Plot the responses to compare the filter outputs obtained with `filter`

and `filtfilt`

.

yzp = filtfilt(Hd,x); figure plot(t(1:100),xin(1:100)) hold on plot(t(1:100),yout(1:100)) plot(t(1:100),yzp(1:100)) xlabel('Time (s)') ylabel('Amplitude') legend('100-Hz Sine Wave','Filtered Signal','Zero-phase Filtering',... 'Location','NorthEast')

In the preceding figure, you can see that the output of `filtfilt`

does not exhibit the delay due to the phase response of the FIR filter.

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