Question

Using filterDesigner in MATLAB, design a second order low pass IIR Butterworth filter whose sampling frequency (Fs) is 1 kHz and cutoff frequency (Fc) is 10 Hz.

Find the numerator and denominator coefficients.

Write its transfer function H(z) = Y(z) / X(z).

Write its difference function y(k).

Draw (copy from Filter Designer) the magnitude response plot.

Draw (copy from Filter Designer) the phase response plot.

Draw (copy from Filter Designer) the impulse response plot.

Answer 1

Magnitude Response (dB) -20 -40 2 -60 -80 100 400 300 2 200 100 0 Frequency (Hz)

Phase Response 0.5 a -2.5 0 100 200 Frequency (Hz) 300 2 400

Impulse Response 0.03 0.025 0.02 0.015 0.01 0.005 150 50 100 200 Time (ms)

File Edit Analysis View Filter Coefficients umera Structure: Direct-Form II Second-Order Sections Order 1.911197067426073203932901378720998764038 0.914975834801433740572917940880870446563 Sections 1 Stable Yes ain 000944691843840150748297379568185760945 Source: Designed t Gain: Store Filter Filter Manager. Filter Order Frequency Specifications Response Type Magnitude Specifications Units: Hz . Specify order: Lowpass ▼ O Highpass Bandpass Fs: 1000 Minimum order The attenuation at cutoff frequencies is fixed at 3 dB the passband power) Fc: 10 Options ○ Bandstop O Differentiator There are no optional parameters for this design method. Design Method IIR Butterworth OFIR Equiripple Design Filter Loading filter. Done

Filter Coefficients Numerator Denominator 1.911197067426073203932901378720998764038 0.914975834801433740572917940880870446563 Gain 0.000944691843840150748297379568185760945 Output Gain

File Edit Analysis View DQMagnitude and Phase Responses Magnitude Response (dB) and Phase Response 0.024 Structure: Direct-Form II Second-Order20 Sections 0.604 1.184 O 40 Order Sections 1 2 -60 1.763 D Stable Yes 2.343 Source: Designed 100 2.922 Store Filter 300 100 200 400 Frequency (Hz) Filter Manager Frequency Specifications Filter Order Response Type Magnitude Specifications Units: Hz FRI Lowpass Highpass Bandpass OSpecify order 2 Fs: 1000 Minimum order The attenuation at cutoff frequencies is fixed at 3 dB the passband power) Fc: 10 Bandstop Options O Differentiator There are no optional parameters for this design method. Design Method OIIR Butterworth FIR Equiripple Design Filter Loading filter. Done

ceef Nr -1 X(2) 1-1.alR +0,CIz Noto dubt eance equa.tion

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