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Two-minute miniquiz problem
Problem 6-1
Given
$H(s)=\frac{s}{(s+1)(s+\text{10})}$
Determine the magnitude of the frequency response in the form of a Bode diagram.
Solution
$H(\mathrm{j\omega})=\frac{1}{\text{10}}\left[\frac{\mathrm{j\omega}}{(\mathrm{j\omega}+1)(\mathrm{j\omega}/\text{10}+1)}\right]$
V. SIGNAL PROCESSING WITH FILTERS
1/ Separation of narrowband signals
The input consists of a sum of two sinusoids, more generally the sum of two narrow-band signals such as the signals transmitted by two radio stations. The objective is to devise filters
to separate these two signals. All waveforms have normalized amplitudes.
2/ Extraction of narrow-band signal from wide-band noise
The input consists of a sinusoid, more generally a narrow-band signal, plus some wide-band noise. The objective is to extract the signal from the noise. All waveforms have normalized amplitudes.
3/ Reduction of narrow-band noise from wide-band signal
The input consists of a wide-band signal, in this case an ecg signal recorded from the surface of the chest, plus some narrowband noise, such as pickup from the power lines. The objective is to remove the narrow-band noise. All waveforms have normalized amplitudes.
VI. LOWPASS AND HIGHPASS FILTERS
1/ First-order lowpass and highpass filters
where H(s) = Y (s)/X(s) and RC = 1.
2/ Use of first-order lpf and hpf for signal separation
How much unwanted signal occurs in each output channel? That depends on the frequency separation between the signals. If the two signals are about a decade apart then the attenuation of the unwanted signal will be at most 20 dB with first-order lowpass and highpass filters.
3/ Higher-order lowpass filters, Butterworth filters
If two signals have a small frequency separation or if attenuation of the unwanted signal needs to be very large, higher-order filters are required. The pole-zero diagrams of the class of lowpass Butterworth filters of order 1-9 are shown below.
The system function for the nth-order Butterworth filter, ${H}_{n}(s)$ , is obtained from the property
${H}_{n}(s){H}_{n}(-s)=\frac{1}{1+{\left[\frac{s}{{\mathrm{j\omega}}_{c}}\right]}^{\mathrm{2n}}}$
where ${\omega}_{c}$ is the cut-off frequency.
Each additional order of Butterworth filter adds an additional attenuation of −20 dB/decade. The frequency response for the nth-order Butterworth filter, ${H}_{n}(\mathrm{j\omega})$ , is obtained from the property
$(\mid {H}_{n}(\mathrm{j\omega})\mid {)}^{2}=\frac{1}{1+{\left[\frac{\omega}{{\omega}_{c}}\right]}^{\mathrm{2n}}}$
The frequency responses are shown for n in the range 1-9 and for ${\omega}_{c}\text{=}1$ .
VII. RESONANCE AND BANDPASS FILTERS
1/ Resonant systems arise in many physical contexts
M is mass, B is a friction constant, K is a spring constant, f(t) is an external force, and v(t) is the velocity of the mass.
A second-order system function relates the velocity to the force
$H(s)=\frac{V(s)}{F(s)}=\frac{\frac{1}{M}s}{{s}^{2}+\frac{B}{M}s+\frac{K}{M}}$
Such a mechanical system yields a damped oscillation in response to a force provided the damping is not too large. For example, a tuning fork may be modeled by such a mechanical system.
2/ RLC filter
Electric networks also show a similar system function. The impedance of the RLC circuit is also of second-order
$Z(s)=\frac{V(s)}{I(s)}=\frac{1}{\text{sC}+\frac{1}{R}+\frac{1}{\text{sL}}}=\frac{\text{sL}}{{s}^{2}\text{LC}+s\frac{L}{R}+1}$
3/ Resonance parameterized
We can parameterize these and any second-order systems efficiently. We illustrate with the electrical network.
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