A system has transfer function .

Determine the sinusoidal steady state output if the input is:

- x(t)=sin(0.1·t)
- x(t)=cos(t+45°)
- x(t)=3·sin(10·t-15°)
- x(t)=4·sin(100·t)

- Does the system exhibit high pass or low pass behavior (i.e., does it preferentially transmit high or low frequency inputs)?

a) H=0.01∠89° (see below). Output is 0.01·sin(0.1·t+89°).

>> H=0.1j/(10+0.1j) H = 0.0001 + 0.0100i >> abs(H) ans = 0.0100 >> angle(H)*180/pi ans = 89.4271

b) H=0.1∠84° (see below). Output is 0.1·cos(t+129°).

>> H=1j/(10+1j) H = 0.0099 + 0.0990i >> abs(H) ans = 0.0995 >> angle(H)*180/pi ans = 84.2894

c) H=0.71∠45° (see below). Output is 2.1·sin(10·t+30°).

>> H=10j/(10+10j) H = 0.5000 + 0.5000i >> abs(H) ans = 0.7071 >> angle(H)*180/pi ans = 45

d) H=1∠6° (see below). Output is 4·sin(100·t+6°).

>> H=100j/(10+100j) H = 0.9901 + 0.0990i >> abs(H) ans = 0.9950 >> angle(H)*180/pi ans = 5.7106

e) The high frequency inputs are transmitted without attenuation. This is a high pass system.

Consider the transfer function .

- Plot the asymptotic (straight line approximations) magnitude and phase plots for a Bode diagram. Assume 3.2 is halfway between 1 and 10 on a log scale. Suitable paper is here.
- Using the Bode plot estimate the sinusoidal steady state output if the input is cos(5t).
- Verify by calculating H(j5). Note: Matlab's "freqresp" command does this, or you can evaluate with your calculator or by hand).
- Use Matlab to plot the Bode diagram using the "bode" command. Use the "grid" command to add a grid (useful for the next part).
- Transfer your asymptotic approximation onto the Matlab plot.

a)

Constant=31 Magnitude=31≈30dB, phase =
0°

Zero at the s=-10 (i.e., break frequency is 10 rad/sec). Slope is
+20 dB/dec, phase goes from 0 to 90°

Pole at s=-1, and at s=-3.2 (i.e., break frequencies at 1 and 3.2 rad/sec).
Slope from each is -20 db/dec, and phase goes from 0 to -90°

b) At ω=5, H(jω)≈12 dB→10^{12/20}=4.0,
∠H(jω)≈-100° Output=4.0cos(5t-100°).

c) Output=3.7cos(5t-109°);

>> H=tf(10*[1 10],[1 4.2 3.2]); >> H_5rad=freqresp(H,5) H_5rad = -1.2333 - 3.4816i >> abs(H_5rad) ans = 3.6936 >> angle(H_5rad)*180/pi ans = -109.5058

d) >> bode(H); grid

e) See part a.

- Sketch (by hand) the Bode plot for . Suitable paper is here.
- Verify with Matlab.

a)

Constant=-0.025 Magnitude=0.025=-32dB, phase =
±180°

Zero at the origin (slope = +20dB/dec, phase=90°)

Double pole at s=-2, slope = -40dB/dec, phase goes from 0 to -180°.

b) >> H=tf(-0.1*[1 0],[1 4 4]); bode(H); grid

The transfer function

represents a low pass transfer function. It's Bode plot is given below.

It seems reasonable to assume that we could make a high pass transfer function by subtracting a low pass transfer function from 1.

As ω→0 H_{LP}(jω)=1,
so H_{HP}(jω)=0. As ω→∞ H_{LP}(jω)=0,
so H_{HP}(jω)=1.

a) Write H_{HP}(s) in standard form (i.e., a single numerator and
denominator polynomial).

b) Draw the Bode plot of H_{HP}(s); use the axes given above
showing H_{LP}(s).

c) Sketch the impulse response of H_{LP}(s).

d) On the same axes, sketch the impulse response of H_{HP}(s).

a)

b)

HLP = tf(1,[0.1 1]); HHP = tf([0.1 0],[0.1 1]); bode(HLP,HHP); legend('H_{LP}(s)','H_{HP}(s)'); grid

c) See part d for graph.

d)

t=linspace(-0.1,0.7,1000); hlp=10*exp(-10*t).*(t>=0); plot(t,hlp,'b','Linewidth',2); hold on; % Plotting highpass is more involved because it involves an impulse. plot(t,-hlp,'r:','Linewidth',2); % Plot exponential stem(0,1,'r:^','Linewidth',2,'Markerfacecolor','r'); legend('h_{LP}(t)','h_{HP}(t)'); grid; hold off axis([-0.1 0.7 -11 11])

It is possible to encode the value of an analog quantity
within a digital signal using a technique called Pulse Width Modulation. With this technique,
the amplitude of the analog quantity is encoded in the *width* of the digital signal.
This problem will show how the amplitude of the original signal can be
determined from the pulse width modulated signal.

The diagram below shows a pulse train, y_{p}(t), of height A,
period T, and pulse width T_{p}. In this case the duty cycle is 40% (T_{p}/T=0.4).
As the analog quantity varies, T_{p} (but not T) will vary.
To represent a quantity of 0.1, the width of the signal is T_{p}=0.1T...

a) Show that the Fourier Series coefficients of y_{p}(t) are given by
.
You can start with the fact that for a single pulse
. This requires no
integration.

Since the function is even the c_{n} are real we can represent y_{p(}t)
by a Fourier cosine series

We use this signal as input to a system with transfer function

b) Sketch the magnitude Bode Plot of H(s) and explain why, if
ω_{LP}
<< ω_{0}, the output will be approximately equal to the
average value of the input a_{0}=AT_{p}/T. (What happens to
signals at frequency ω_{0} or higher?)

c) In particular show that the constant term (a_{0}) is passed by
the system without attenuation (what is the gain of the system as frequency
goes to 0?).

d) Show that the magnitude of the first harmonic, a_{1}, is
maximized when T_{p}=T/2 (i.e., the duty cycle is 50%). Since
the first harmonic is the larges of all the harmonics, and is attenuated the
least by the system (it is the lowest frequency), this should be the worst
case scenario in terms of variations, or ripples, in the output. This
requires no differentiation.

e) If T_{p}=T/2 what should the value of ω_{0}be in relation to ω_{LP}
such
that the output at the frequency of the first harmonic is only 1% of the
output due to the DC component (i.e., a_{0}). The output of
the system will then be almost constant with an amplitude determined by the
pulse width.

a)

b) If ω_{0}>>ω_{LP}
then all the harmonics of the series will be greatly attenuated leaving only
the constant component (a_{0}) which is not changed since the gain
is 0 dB (quantity=1) as ω→0.

c) Since H(0)=1, the constant signal (frequency=0) passes without attenuation.

d)

This is maximized when
or T_{p}=T/2

e) The magnitude of the first harmonic (in the output) is the
magnitude of the harmonic (frequency=ω_{0})
times the magnitude of the transfer function evaluated at jω_{0}.

We want to find the value of
such that this 1% of a_{0}=A/2.

a) Find the differential equation for the system with Bode plot shown.

b) Show a system of your own design that has the given Bode plot.

c) What are ζ and ω_{0}?

a,c) DC gain=1 (0 dB) and there is a double pole at w0=10 rad/sec (mag drops 40 dB/dec, and phase goes from 0 to -180°.

b)