Final Review Signals

Question 1

Expression for an energy signal

Answer 1

E = integral from -infinity to +infinity (x^2(t)) dt

Question 2

where does t of the energy limit tend to

Answer 2

0

Question 3

if t in the energy limit tends to infitity, solve for

Answer 3

power

Question 4

expression for a power signal

Answer 4

P = lim t->1/T * infinity of an integral from 0 to t |x^2(t)| dt

Question 5

characteristic of discrete signals

Answer 5

has a value for only certain moments in time

Question 6

characteristic of continuous signals

Answer 6

has a value for all moments in time

Question 7

characteristic of analogue signals

Answer 7

has an amplitude at any time

Question 8

characteristic of digital signals

Answer 8

finite amplitudes (square wave)

Question 9

characteristic of periodic signals

Answer 9

signal pattern repeats for all time

Question 10

characteristic of non/aperiodic signals

Answer 10

does not repeat

Question 11

trig fourier series

Answer 11

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

Question 12

complex fourier series

Answer 12

g(t) = sum(C*e^(jnwt))

Question 13

fourier transform

Answer 13

G(w) = integral from -infinity to infinity (g(t)e^(-jwt))dt

Question 14

discrete time fourier transform

Answer 14

x(k) = sum(XnWn^(kn))

Question 15

relationship between the coefficients of complex and trig fourier series

Answer 15

complex = Xn
trig = An, Bn

relationship: Xn= (An-jBn)/2

Question 16

fundamental frequency equation

Answer 16

w = 2pi/T

T = period

Question 17

If a signal can be decomposed as a linear combination of sinusoids and co-sinusoids, is it certain to result in a periodic waveform

Answer 17

NO

combination of sinusoinds and co-sinusoides can create nonperiodic functions

dependent of the ratio of frequencies

Question 18

duality

Answer 18

when signals are inverses of each other

Question 19

sifting theorem

Answer 19

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

Question 20

fourier transform of unit impulse

Answer 20

F(w) = integral from -infinity to infinity (d-delta(t)e^-jwt == 1

Question 21

fourier transform of impulse train

Answer 21

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

Question 22

Parsevals Theorem

Answer 22

integral from -infinity to infinity( |x^2(t)|)dt = 1/2pi * integral from -infinity to infinity (G^2(w)) dw

Question 23

fourier transform of g(t) = x(t − τ )

Answer 23

time shift -> frequency shift

from reference table:
e^-jwτ G(w)

Question 24

fourier transform of h(t) = x(t)* e^jωt

Answer 24

multiplied by a complex phasor -> modulation -> frequency shift

from reference table:
G(w-w0)

Question 25

when a shift in time occurs, there is a

Answer 25

multiplication by a complex phasor in the frequency domain

Question 26

to represent the linear combination of a fourier transform

Answer 26

substitue variables into Fourier transform

replace, reorder, recognize

Question 27

practical sampled data system

Answer 27

x(t) -> anti-aliasing filter -> ADC [ sample/hold -> quantizer] -> DSP [processor] -> DAC [analogue converter] -> recon. filer [low pass filter]

Question 28

pros/cons of low pass filter

Answer 28

pros: avoids distortion

cons: lose info, large # of components needed to make, expensive

Question 29

Nyquist’s sampling theorem

Answer 29

keeps all the information in the signal when you sample it

have to sample it twice the max frequency within the signal

Question 30

how does the spectrum of a sampled signal relate to the spectrum of the original continuous-time signal?

Answer 30

sampled spectrum = original spectrum + spectral images

Question 31

what are spectral images

Answer 31

images of the spectrum shifted by integer multiples of the sampling frequency

Question 32

what does a reconstruction filter do

Answer 32

eliminates aliases, reconstructs an analogue signal from its digital representation

Question 33

explain why, in practical analogue-to-digital and digital-to-analogue conversion applications, it is impractical to sample a signal perfectly.

Answer 33

noise, distortion, and quantization

Question 34

Why is perfect signal
reconstruction impossible even when a perfectly sampled signal is available?

Answer 34

always going to be imperfections in the reconstruction process

Question 35

non-periodic signal that can be classified as a power signal

Answer 35

impulse

Question 36

main subsystems to analogue to digital block

Answer 36

sampling, quantization, coding

Question 37

main subsystems to digital to analogue

Answer 37

interpolation, reconstruction

Question 38

in either analogue->digital conversion or digitial->analogue conversion, why is it impractical to sample a signal perfectly

Answer 38

need an infinitely high sampling rate, limitations in components, quantization error, system constrains, and cost

Question 39

power efficiency formula

Answer 39

useful power / total power

Question 40

Nyquist frequency formula

Answer 40

1/2* sampling rate

Question 41

Nyquist rate formula

Answer 41

2*bandwidth

Question 42

definition of a fourier transform

Answer 42

function derived from a given function, represented by sinusoidal functions