Final Review Signals Flashcards
Expression for an energy signal
E = integral from -infinity to +infinity (x^2(t)) dt
where does t of the energy limit tend to
0
if t in the energy limit tends to infitity, solve for
power
expression for a power signal
P = lim t->1/T * infinity of an integral from 0 to t |x^2(t)| dt
characteristic of discrete signals
has a value for only certain moments in time
characteristic of continuous signals
has a value for all moments in time
characteristic of analogue signals
has an amplitude at any time
characteristic of digital signals
finite amplitudes (square wave)
characteristic of periodic signals
signal pattern repeats for all time
characteristic of non/aperiodic signals
does not repeat
trig fourier series
an = 2/T * integral from -T/2 to T/2 (g(t)cos(nwt))dt
bn = 2/T * integral from -T/2 to T/2 (g(t)sin(nwt))dt
complex fourier series
g(t) = sum(C*e^(jnwt))
fourier transform
G(w) = integral from -infinity to infinity (g(t)e^(-jwt))dt
discrete time fourier transform
x(k) = sum(XnWn^(kn))
relationship between the coefficients of complex and trig fourier series
complex = Xn
trig = An, Bn
relationship: Xn= (An-jBn)/2
fundamental frequency equation
w = 2pi/T
T = period
If a signal can be decomposed as a linear combination of sinusoids and co-sinusoids, is it certain to result in a periodic waveform
NO
combination of sinusoinds and co-sinusoides can create nonperiodic functions
dependent of the ratio of frequencies
duality
when signals are inverses of each other
sifting theorem
if you multiply a signal by an impulse and integrate over all time, you get a signal evaluated at the time position aka. the sample of a signal
integral from -infinity to infinity(x(t)d-delta(t-Tau)dt = X(Tau) == integral from -infinity to infinity (signalimpulse) = sample
fourier transform of unit impulse
F(w) = integral from -infinity to infinity (d-delta(t)e^-jwt == 1
fourier transform of impulse train
s(t) = sum(d-delta(t-nT) = 1/T integral from -T/2to T/2 (d-delta(t)e^-jnwt dt == 1/T (e^-jnwt) evluated at t=0 === 1/T
Parsevals Theorem
integral from -infinity to infinity( |x^2(t)|)dt = 1/2pi * integral from -infinity to infinity (G^2(w)) dw
fourier transform of g(t) = x(t − τ )
time shift -> frequency shift
from reference table:
e^-jwτ G(w)
fourier transform of h(t) = x(t)* e^jωt
multiplied by a complex phasor -> modulation -> frequency shift
from reference table:
G(w-w0)
when a shift in time occurs, there is a
multiplication by a complex phasor in the frequency domain
to represent the linear combination of a fourier transform
substitue variables into Fourier transform
replace, reorder, recognize
practical sampled data system
x(t) -> anti-aliasing filter -> ADC [ sample/hold -> quantizer] -> DSP [processor] -> DAC [analogue converter] -> recon. filer [low pass filter]
pros/cons of low pass filter
pros: avoids distortion
cons: lose info, large # of components needed to make, expensive
Nyquist’s sampling theorem
keeps all the information in the signal when you sample it
have to sample it twice the max frequency within the signal
how does the spectrum of a sampled signal relate to the spectrum of the original continuous-time signal?
sampled spectrum = original spectrum + spectral images
what are spectral images
images of the spectrum shifted by integer multiples of the sampling frequency
what does a reconstruction filter do
eliminates aliases, reconstructs an analogue signal from its digital representation
explain why, in practical analogue-to-digital and digital-to-analogue conversion applications, it is impractical to sample a signal perfectly.
noise, distortion, and quantization
Why is perfect signal
reconstruction impossible even when a perfectly sampled signal is available?
always going to be imperfections in the reconstruction process
non-periodic signal that can be classified as a power signal
impulse
main subsystems to analogue to digital block
sampling, quantization, coding
main subsystems to digital to analogue
interpolation, reconstruction
in either analogue->digital conversion or digitial->analogue conversion, why is it impractical to sample a signal perfectly
need an infinitely high sampling rate, limitations in components, quantization error, system constrains, and cost
power efficiency formula
useful power / total power
Nyquist frequency formula
1/2* sampling rate
Nyquist rate formula
2*bandwidth
definition of a fourier transform
function derived from a given function, represented by sinusoidal functions
