2P6 Information Engineering Flashcards
How do you find the energy of a signal?
Parsaval’s theorem: integral(|x(t)|^2) over all real numbers
How do you find the power of a signal?
P = lim T->inf (1/T integral(|x(t)|^2)) from -T/2 to +T/2
Define bandwidth
The range of frequencies over which its spectrum is non-zero. |X(f)| where X(f) is the fourier transform of x(t)
What is a passband signal?
A signal where its spectral content is centered around +- fc,
Usually fc+W to fc-W. so the bandwidth is 2W.
What is rect(t/T)?
Rectangular pulse, which =1 for -T/2 <t<T/2
What are some alternative definitions of frequency which are more realistic?
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
How can you model a channel?
y(t) = h(t)*x(t) + n(t)
How is noise in a channel modelled?
As a Gaussian random process, at each time t, n(t) is a Gaussian random variable.
For normal AM modulation what is the moduluation index limited to?
mA = max(x(t))/a0
mA <1 because otherwise there will be phase reversal.
What is a reciever for a normal AM receiver?
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
What is the spectrum of an ordinary AM sigal?
delta function at carrier frequency, and the information bandwidth shifted to symetrically around the carrier frequency.
What is the power of an ordinary AM signal withoriginal power Px, and offset a0?
a0^2/2 + Px/2
What is DSB-SC?
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)
How does a DSB-SC reciever work?
modulator + lowpass filter
s(t) * cos(2pifct)
What is SSB-SC?
Only need to specify X(f) for f>0
What is the power of of SSB-SC and DSB-SC?
DSB-SC -> Px/2
SSB-SC -> Px/4
What is the expression for an FM modulated signal?
s(t) = Ac cos [ 2pi fc t + 2pi kf integral(x(u)du)]
because x determines instantaenous frequency, so needs to be integrated.
How do you obtain the original signal from an FM signal?
differentiate giving passband signal with AM, then can use envelope detection
differentiator + envelope detector.
Why is FM more robust than AM?
message is hidden infrqeucny rather than amplitude, which is less affected by noise. But this leads to increased bandwidth.
What is the moduluation index of an FM signal?
Δf/fx, where Δf is the max deviation of teh carrier frequency f(t) from fx.
What is the spectrum of an FM wave?
Bessel function multipled by delta functions either side of fc and -fc.
What is the effective bandwidth of an FM signal?
Bfm = 2Δf + 2fx
What’s the difference between sampling and Quantisation?
Sampling discretises the time axis.
Quantisation discretises the signal amplitude axis
What is the fourier transform of the sampled signal?
Xs(f) = 1/T * Sum (X(f-nfs))
What is uniform quantisation?
Finite set of levels, even step between each level, assign bits to each level.
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.
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]
What is noise power for a uniformlly distributed noise with a step size of Δ?
Δ^2/12
RMS noise = Δ/root(12)
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)
What is the rate of a digitised uniform quantiser?
n2W (bits/sec)
bandwidth W, n-bit uniform quantiser
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
What is a constellation?
The set of value the bits are mapped to
In a constellation with M symbol, each symbol represents how many bits?
log2(M)
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
What shapes satisfy the orthonormal shift property?
sinc and rect
What is a common pulse shape?
root raised cosine spectrum, quick decay but larger bandwidth
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
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.
How does does probability of detection error depend on snr?
Pe = e^(-snr/2)
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)
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
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.
Where are the decision regions for QAM?
Bisextors inbetween the points.
What happens when you increase the number of constellation poins with QAM?
the transmission rate increases,
however, the power increases
What is FSK?
Frequency shift keying, shifting the frequecny for each symbol of constellation.
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.
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
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.
What is TDMA,FDMA,CDMA?
TDMA - Time division multiple access
FDMA - frequency division multiple access
CDMA - code division multiple access
How to demodulate FDMA?
modulate with different carrier frequencies depending on which signal you want to extract.
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.
