Bandpass Filter Response

This example shows how to compute the time-domain response of a simple bandpass filter:

  1. Choose inductance and capacitance values using the classic image parameter design method.

  2. Use circuit, capacitor, and inductor with the add function to programmatically construct a Butterworth circuit.

  3. Use setports to define the circuit as a 2-port network.

  4. Use sparameters to extract the S-parameters of the 2-port network over a wide frequency range.

  5. Use s2tf to compute the voltage transfer function from the input to the output.

  6. Use rationalfit to generate rational fits that capture the ideal RC circuit to a very high degree of accuracy.

  7. Create a noisy input voltage waveform.

  8. Use timeresp to compute the transient response to a noisy input voltage waveform.

Design a Bandpass Filter by Image Parameters

The image parameter design method is a framework for analytically computing the values of the series and parallel components in passive filters. For more information on this method, see "Complete Wireless Design" by Cotter W. Sayre, McGraw-Hill 2008 p. 331.

Figure 1: A Butterworth bandpass filter built out of two half-sections.

The following MATLAB code generates component values for a bandpass filter with a lower 3-dB cutoff frequency of 2.4 GHz and an upper 3-dB cutoff frequency of 2.5 GHz.

Ro = 50;
f1C = 2400e6;
f2C = 2500e6;

Ls = (Ro / (pi*(f2C - f1C)))/2;
Cs = 2*(f2C - f1C)/(4*pi*Ro*f2C*f1C);

Lp = 2*Ro*(f2C - f1C)/(4*pi*f2C*f1C);
Cp = (1/(pi*Ro*(f2C - f1C)))/2;

Programmatically Construct the Circuit

Before building the circuit using inductor and capacitor objects, we must number the nodes of the circuit shown in figure 1.

Figure 2: Node numbers added to the Butterworth bandpass filter.

Create a circuit object and populate it with inductor and capacitor objects using the add function.

ckt = circuit('butterworthBPF');

add(ckt,[3 2],inductor(Ls))
add(ckt,[4 3],capacitor(Cs))
add(ckt,[5 4],capacitor(Cs))
add(ckt,[6 5],inductor(Ls))

add(ckt,[4 1],capacitor(Cp))
add(ckt,[4 1],inductor(Lp))
add(ckt,[4 1],inductor(Lp))
add(ckt,[4 1],capacitor(Cp))

Extract S-Parameters From the 2-Port Network

To extract S-parameters from the circuit object, first use the setports function to define the circuit as a 2-port network. Once the circuit has ports, use sparameters to extract the S-parameters at the frequencies of interest.

freq = linspace(2e9,3e9,101);

setports(ckt,[2 1],[6 1])
S = sparameters(ckt,freq);

Fit the Transfer Function of the Circuit to a Rational Function

Use the s2tf function to generate a transfer function from the S-parameter object. Then use rationalfit to fit the transfer function data to a rational function.

tfS = s2tf(S);
fit = rationalfit(freq,tfS);

Verify the Rational Fit Approximation

Use the freqresp function to verify that the rational fit approximation has reasonable behavior outside both sides of the fitted frequency range.

widerFreqs = linspace(2e8,5e9,1001);
resp = freqresp(fit,widerFreqs);

xlabel('Frequency (Hz)')
title('The rational fit behaves well outside the fitted frequency range.')

Construct an Input Signal to Test the Band Pass Filter

This bandpass filter should be able to recover a sinusoidal signal at 2.45 GHz that is made noisy by the inclusion of zero-mean random noise and a blocker at 2.35 GHz. The following MATLAB code constructs such a signal from 4096 samples.

fCenter = 2.45e9;
fBlocker = 2.35e9;
period = 1/fCenter;
sampleTime = period/16;
signalLen = 8192;
t = (0:signalLen-1)'*sampleTime; % 256 periods

input = sin(2*pi*fCenter*t);     % Clean input signal
noise = randn(size(t)) + sin(2*pi*fBlocker*t);
noisyInput = input + noise;      % Noisy input signal

Compute the Transient Response to the Input Signal

The timeresp function computes the analytic solution to the state-space equations defined by the rational fit and the input signal.

output = timeresp(fit,noisyInput,sampleTime);

View Input Signal and Filter Response in the Time Domain

Plot the input signal, noisy input signal, and the band pass filter output in a figure window.

xmax = t(end)/8;
axis([0 xmax -1.5 1.5])

axis([0 xmax floor(min(noisyInput)) ceil(max(noisyInput))])
title('Noisy Input')
ylabel('Amplitude (volts)')

axis([0 xmax -1.5 1.5])
title('Filter Output')
xlabel('Time (sec)')

View Input Signal and Filter Response in the Frequency Domain

Overlaying the noisy input and the filter response in the frequency domain explains why the filtering operation is successful. Both the blocker signal at 2.35 GHz and much of the noise is significantly attenuated.

NFFT = 2^nextpow2(signalLen); % Next power of 2 from length of y
Y = fft(noisyInput,NFFT)/signalLen;
samplingFreq = 1/sampleTime;
f = samplingFreq/2*linspace(0,1,NFFT/2+1)';
O = fft(output,NFFT)/signalLen;

axis([freq(1) freq(end) 0 1.1])
legend('filter transfer function')

axis([freq(1) freq(end) 0 1.1])
title('Filter characteristic and noisy input spectrum.')
xlabel('Frequency (Hz)')
ylabel('Magnitude (Volts)')

For an example of how to compute and display this bandpass filter response using RFCKT objects, go to: Bandpass Filter Response Using RFCKT Objects.