1
Q
How do you find the energy of a signal?
A
Parsaval’s theorem: integral(|x(t)|^2) over all real numbers
2
Q
How do you find the power of a signal?
A
P = lim T->inf (1/T integral(|x(t)|^2)) from -T/2 to +T/2
3
Q
Define bandwidth
A
The range of frequencies over which its spectrum is non-zero. |X(f)| where X(f) is the fourier transform of x(t)
4
Q
What is a passband signal?
A
A signal where its spectral content is centered around +- fc,
Usually fc+W to fc-W. so the bandwidth is 2W.
5
Q
What is rect(t/T)?
A
Rectangular pulse, which =1 for -T/2 <t<T/2
6
Q
What are some alternative definitions of bandwith which are more realistic?
A
90% bandwidth, the range of frequencies which contain 90% of the energy of the spectrum
3- dB bandwith, range of frequencies that contain 50% o the energy of teh spectrum.
null->null bandwidth, the width of the main lobe
7
Q
How can you model a channel?
A
y(t) = h(t)*x(t) + n(t)
8
Q
How is noise in a channel modelled?
A
As a Gaussian random process, at each time t, n(t) is a Gaussian random variable.
9
Q
For normal AM modulation what is the moduluation index limited to?
A
mA = max(x(t))/a0
mA <1 because otherwise there will be phase reversal.
10
Q
What is a reciever for a normal AM receiver?
A
diode into a capacitor and resistor in parallel.
Positive half cycle, capacitor chargers up rapidly, when input falls below, diode is reverse biased so slowly dischargers through Rl
11
Q
What is the spectrum of an ordinary AM sigal?
A
delta function at carrier frequency, and the information bandwidth shifted to symetrically around the carrier frequency.
12
Q
What is the power of an ordinary AM signal withoriginal power Px, and offset a0?
A
a0^2/2 + Px/2
13
Q
What is DSB-SC?
A
Double Side Band Suppressed Carrier,
get rid of a0, to get rid of delta function at fc,
s(t) = x(t) cos(2pi fc t)
14
Q
How does a DSB-SC reciever work?
A
modulator + lowpass filter
s(t) * cos(2pifct)
15
Q
What is SSB-SC?
A
Only need to specify X(f) for f>0
16
Q
What is the power of of SSB-SC and DSB-SC?
A
DSB-SC -> Px/2
SSB-SC -> Px/4
17
Q
What is the expression for an FM modulated signal?
A
s(t) = Ac cos [ 2pi fc t + 2pi kf integral(x(u)du)]
because x determines instantaenous frequency, so needs to be integrated.
18
Q
How do you obtain the original signal from an FM signal?
A
differentiate giving passband signal with AM, then can use envelope detection
differentiator + envelope detector.
19
Q
Why is FM more robust than AM?
A
message is hidden infrqeucny rather than amplitude, which is less affected by noise. But this leads to increased bandwidth.
20
Q
What is the moduluation index of an FM signal?
A
Δf/fx, where Δf is the max deviation of teh carrier frequency f(t) from fx.
21
Q
What is the spectrum of an FM wave?
A
Bessel function multipled by delta functions either side of fc and -fc.
22
Q
What is the effective bandwidth of an FM signal?
A
Bfm = 2Δf + 2fx
23
Q
What’s the difference between sampling and Quantisation?
A
Sampling discretises the time axis.
Quantisation discretises the signal amplitude axis
24
Q
What is the fourier transform of the sampled signal?
A
Xs(f) = 1/T * Sum (X(f-nfs))
25
What is uniform quantisation?
Finite set of levels, even step between each level, assign bits to each level.
26
Loss rate of sampling and quantisation?
Sampling is a lossless procedure as long as the sampling rate is higher than the Nyquist rate.
Quatisation is always lossy.
27
Define quantisation noise?
Q(x(nT)) is the quantisation of x(nT)
then
eQ = x(nT) - Q(x(nT))
if step size is Δ
then eq(z) ε [-Δ/2 , +Δ/2]
28
What is noise power for a uniformlly distributed noise with a step size of Δ?
Δ^2/12
RMS noise = Δ/root(12)
29
What is signal to noise ratio?
signal power/noise power = (RMS)^2/(RMS)^2
= (V^2/2) / (Δ^2/12) = 3 x 2^(2n-1)
30
What is the rate of a digitised uniform quantiser?
n2W (bits/sec)
bandwidth W, n-bit uniform quantiser
31
How can we reduce the bit-rate of signals by using quantisation?
non-uniform step size, more steps at smaller values, less steps at larger (rarer) values
32
What is a constellation?
The set of value the bits are mapped to
33
In a constellation with M symbol, each symbol represents how many bits?
log2(M)
34
What is the orthonomal shifts property of digital pulses?
integral(p(t-kT)p(t-mT)dt) is only equal to 1 when k=m
35
What shapes satisfy the orthonormal shift property?
sinc and rect
36
What is a common pulse shape?
root raised cosine spectrum, quick decay but larger bandwidth
37
What is a matched filter?
y(t) is passed through filter with impulse response p(-t)
the filter output is then xb(t) * h(t) = sum Xk integral(p(τ-kT)p(τ-t)dτ)
which is sampled only at t=mT so you only get Xm out
38
What happens when there is noise at the reciever of a matched filter?
you get r(mT) = Xm + Nm
where Nm = integral(n(τ)p(τ-mT)dτ)
decoder usually picks te most likely one.
39
How does does probability of detection error depend on snr?
Pe = e^(-snr/2)
40
What is the difference between digital baseband and digital passband communications?
Passband -> the baseband signal is moduluated by a high frequency carrier.
x(t) = sum(Xk p(t-kT)) cost(2pifc t)
41
What is quadrature amplitude modulation?
make the constellations complex valued
x(t) = Re[xb(t)e ^(j2pifct)]
the cosine component carriers the real part of the signal and the sine component carriers the imaginary part of the signal
42
How is QAM demodulated?
By using two seperate de modulators.
One multiples by a cos and the other by a -sin,
they then end up seperately with the real and imaginary parts of the original signal. and the constellation can be extracted.
43
Where are the decision regions for QAM?
Bisextors inbetween the points.
44
What happens when you increase the number of constellation poins with QAM?
the transmission rate increases,
however, the power increases
45
What is FSK?
Frequency shift keying, shifting the frequecny for each symbol of constellation.
46
What are repetition codes?
Repeating each information bit odd number of times, and then picking the majority one. Slow rate 1/n, not hugely improved probability of correct.
47
What is a block coding?
Passing a block of bits larger than actual number of bits, rate>0.5, but keep redundancy up.
first bits are the original, the other are dependent
48
What is Hamming codes?
N=7 K=4, first 4 bits are information, last three are redundant bits.
c5 = s1+s2+s3
c6 = s2 + s3 + s4
c7 = s3 + s4 + s1
where + is moduluo 2 addition.
rate is 4/7.
Can only correct one flipped bit.
49
What is TDMA,FDMA,CDMA?
TDMA - Time division multiple access
FDMA - frequency division multiple access
CDMA - code division multiple access
50
How to demodulate FDMA?
modulate with different carrier frequencies depending on which signal you want to extract.
51
How do you demodulate CDMA?
Integrate with the code that you want to extract, due to orthogonality of the codes, you will only end up with the one you are looking for.
52
How does BPSK?
Binary phase shift keying, to alternate 1s and 0s, phase is flipped 180o, carrier is usually a sine wave.
53
How does the Watt's Governor work?
Slower rotation, balls go in, opens the valve more allowing more steam into engine.
54
How do block diagrams work?
Blocks indicate functions, such as ODE, or amplifier, then inputs are values. Lines represent the flow of information not the flow of 'stuff'
55
What is 'plant' and what is 'controller'?
Plant is fixed blocks which are usually the dynamics of the system.
Whereas controller is designed block to control the system, geometry of flyball is a control block.
56
Define linear system.
f(u1) +f(u2) = f(u1 + u2)
57
What does the assumption 'causal' mean in terms of linear systems?
the output at time T, y(T) depends only on the input up to time t where t<=T. This is a fundemental property of all time-based systems.
58
What does the assumption time-invariant mean?
u(t)-> y(t)
u(t-T) -> y(t-T)
59
What are the roots and poles of a transfer function G(s)?
zeros at the roots of the numerator.
poles at the roots of the denominator.
60
Final value and intial value theorem?
lim[t->inf](y(t)) = lim[s->0](sy(s))
lim[t->0](y(t)) = lim[s->inf](sy(s))
61
Proof of the Laplace transform of the convolution integral?
Set T = t-τ, fiddle around with the limits, integrate with respect to t firsst.
62
How is step response related to impulse response?
Step response = integral of impulse response
63
Difference between lag and delay?
Lag 1/(s+a),
Delay e^(-sT), y(t)=u(t-T)
64
If G(s) is the transfer function from x -> z, then what is the expression for G?
z = Gx
z/x = G
65
How is the transfer function related to the time domain?
Laplace transform of the impulse response.
66
Define asymptotic stability?
LTI system is aymptotically stable if its impulse response g(t) satisfies: integral[0->inf] (|g(t)|dt)
67
What does it mean for a fraction to be proper?
numerator polynomial has a lower or equal to degree than the denominator.
68
Conditions for stability based on pole locations and explanation for it?
LTI system with rational transfer function G(s) is asymptotically stable if, and only if, all poles of G(s) lie in the LHP.
Can split rational transfer function into a series of partial fractions, if the poles have positive real parts it will lead to exponentially increasing terms,
69
Proof of asymptotic stability.
Pole pk = σk + jωk
each term is exp(σkt) exp( jωk) < exp(σkt)
integral (exp(σkt)) = -1/ σ
=> integral (g(t)) < |a0| + |a1/σ1| + ... < inf
=> stable.
If there are repeated roots you get β1e^pt + β2te^pt + ... + βl/(l-1)! t^(l-1) e^pt
if σ<0, then integral(g(t))
70
Define marginal stability.
integral[0->T](|g(t)|dt) < A+BT
71
What is the full stability theorem for systems with proper rational transfer function?
1. System is asymptotically stable if all its poles have negative real parts
2. System is unstable if any pole has a positive real part, or if there are any repeated poles on the imaginary axis
3. A system is marginally stable if it has one or more distinct on the imaginary axis and any remaining poles have negative real parts.
72
What does real part, imaginary part, magnitude, and arguement of a pole determine?
Real part -> stability and time constant
Imaginary part -> damped natural frequency
Magnitude -> natural frequency
Arguement -> damping ratio.
73
What happens when a pure sinusoid is an input to an aymptotically stable LTI system?
Output will also settle down to a pure sinusoid with the same frequency, but different amplitude and phase.
74
How do you derive the ss response of a sinusoidally excited LTI system?
y(s) = G(s) u(s)
u(t) = exp(jωt), u(s) = 1/(s-jω)
y(s) = G(s)1/(s-jω) = λ1/(s-p1) + λ2/(s-p2) + λn/(s-pn) + λ0/(s-jω)
use cover up, multiply by (s-jω), to see that λ0 = G(jω)
=> Re(y(t)) = Re(exp(jωt)G(jω))
75
What does a bode plot of the transfer function (sT)^k look like?
Straight line, fixed frequency.
k=0, dB=0, φ=0
k=1, +20dB/decade, φ=90
k=-1, -20dB/decade, φ=-90
76
Bode plot of first order (1+sT)
3dB point at ω=1/T, +20dB after this point
Goes +90o through the 1/T
77
Bode plot of G(s) = 1/(1+ 2ςsT + s^2T^2)
in databook,
small hump (or dip) at 1/T
Goes from 0->-180 over 1/T
78
How to plot Bode plot of complex transfer function?
Plot the striaght line element of each part of the function and then add them linearly together.
79
What is the return ratio?
L(s) = H(s)G(s)K(s)
where H(s) is the feedback fraction
K(s) is the controller
G(s) is the plant
80
What is general form of closed loop transfer function?
GK/(1+L) where L = HGK
81
What are advantages of proprtional, derivative and integral action?
proportional -> basic, but hard for good damping and small errors
derivativve -> improves the damping
Integral -> reduces steady state errors
82
What are:
r(s), e(s), u(s), di(s), do(s), y(s)
r(s) ->reference
e(s) -> error signal
u(s) -> control signal
di(s) -> input disturbance
do(s) ->output disturbance
y(s) -> controlled output
83
What are closed-loop poles?
The zeros of 1+HGK =1+L
1+L=0 is the closed-loop characteristic equation.
84
What do the closed loop poles determine?
The stability
Characteristics ofthe system's transient response
85
Define return ratio?
-1 x product of all ters round the loop.
86
What is the open loop response of a system?
Create a break at the - terminal of the summation block. a(s) = -HGKb(s)
This is often done to test before closing the loop.
87
What does H represent in a feedback control system
Sesnor dynamics
88
What is sensitivity S(s) and complementary sensitivity T(s)?
S(s) = 1/(1+L) which is the transfer function from output disturbances
t(s) = L/(1+L) which is the transfer function from the reference to the output.
89
What is the steady state response of a system with a step input U for t>0??
G(0)U
90
What is the steady state error?
1/(1+L(0))
91
What is a root locus diagram?
A diagram which shows how the closed-loop poles move for changing gain, k.
92
Why is the steady state error for integral control 0?
K=Ki/s, K(0) = inf and since e = 1/(1+GK) e->0
If the error tended to a steady state value, then the integrator would tend to infinity, which contradicts the asymptotic stability assumption of the system.
93
What is the Nyquist diagram?
A plot of the frequency response of the return ratio, on an argand diagram.
94
What is the Nyquist stability crierion?
If the open-loop is symptotically stable (L(s) has all roots in the LHP) and the Nyquist diagram does not enclose the point '-1' then the closed loop will be asymptotically stable.
95
What is the Nyquist plot of the integrator?
IT is a single straight line on the -j axis, with ω->inf at G=0
96
What does the Nyquist plot of the Time delay look like?
A circle
97
What does the Nyquist plot of a first order lag look like?
A semi-cricle in the RHP
98
How can you determine limiting behaviour of Nyquist plot?
Take Taylor series of numerator and denominator G(jω) and evaluate at ω=0
99
What is the overall strategy for determining Nyquist plot?
Find asymptotic behaviour, and calculate a few points inbetween.
If G(0) is finite and non-zero then Nyquist locus will always start by leaving real axis at right angles to it. if G(0 is infinite) then take taylor series of G(jω) about ω=0
100
When can oscillation occur, even without external input?
L(jω1) =-1 for some ω1, then L/(1+L) is infinite only at that frequency. Then if e(t) = cos(ω1t) r(t)=0, sustained oscillation.
101
Define gain margin and phase margin.
gain margin is 1/α where -α is the point where the Nyquist locus crosses the real axis.
Phse margin = θ where this is the arguement of the point closes to the real axis which has magnitude of 1.
102
How do you get good feedback properties with sensitivity?
Aim for small sensitivity over appropriate frequncies, i.e. 1/|1+L(jω)| << 1 within the appropriate bandwidth.
103
What is the closed loop bandwidth?
|S(jω1)| =1, |T(jω2)| = 1
where S is the sensitivity (1/(1+L))
and T is the complementary sensitivity
L/(1+L)
bandiwdth is ω1->ω2
between these frequncies both disturbances and reference signals are amplified.
104
How can you find the points where |S(jω)|=1 and |T(jω)|=1 from a Nyquist plot?
|S(jω)|=1 is 1 when |1+L(jω)| =1 which is when the distance from the point -1 to the Nyquist locus = 1.
|T(jω)|=1 when |L(jω)|=|1+L(jω)|
when the distance from point -1 to the Nyquist locus equals the distance from the origin to the Nyquist locus. (always occurs at Re(L) = -0.5)
105
How do you find from the Nyquist plot where the sensitivity is maximized?
When the distance from the point -1 to the Nyquist locus is at minimum, becase |1+L| at minimum
106
How do you find the point where |T(jω)| is maximised?
Try a few points around where|1+L| = minimimum
107
What is the ful Nyqsuit stability theorem for stable L(s)
Feedback system is asymptotically stable if and only if the point -1 if not encircled by the "full" Nyquist diagram of L, for -inf < ω<+inf
Encirclements must be added algebraically, anti clockwise and clockwise encirclements cancel
If L(s) has one or more poles at 0, it still works but if it has n poles, the Nyquist plot should be completed by adding a large nx180 arc in clockwise direction.
If L(s) is unstable and has n unstable poles, then the feedback system is stable if and only if the full Nyquist diagram makes n anti clockwise encirclements of the point -1
More ambigiously, the feedback system is stable if the Nyquist diagram of L leaves the point -1 on its left.
108
What 3 things are desirable in a feedback system?
|KG|>>1 for frequnecy where benefits of feedback are sought. such that S<<1
|KG|<<1 for high frequencies (ω>ωc)
KG satisfies the Nyquist stability criterion, with adequate gain and phase margins, ensuring neither S or T have a large peak in crossover region.
109
What is a phase lag and a phase lead compensator?
Phase lag P+I, K(s) = k (s+α)/(s+β) β<α< ωc
Improves low frequency gain (reducing steady-state errors), at the expensve of adding phase lag between β and α
but if ωc is > β,α then this is ok
Phase lead K(s)= k (s+α)/(s+β) for α<β and α<ωc<β
This decreases low frequency gain but increases high frequnecy gain (usually bad). However, it increases phase around ωc
110
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