Comm Engineering Test 1
What is important about positive and negative frequencies?
negative does not exist, rather it just mirror the positive side
signal power for periodic signals
periodic signals start at - infinity and continue for ever Pg and Px are powers of signal g and signal x
explain time shifting
- is delay (right) + is advance (left) in terms of original signal
what does the magnitude spectrum look like in double sideband amplitude supressed carrier modulation and why?
4 pulses knowing the inital signal had its own sinusoidal frequency so it is like how we had to distribute out the dirac delta for the other problems 4 pulses surrounded on +- fc no pulses actually at fc unless the signal MF, we are trying to modulate has inpulse at 0 frequency
what does the DSBSC modulated signal look like?
Ac m(t) cos wct
DFT formulas written other way (CORRECT inverse DFT)
F = 1/N DFT is basically the discrete time fourier series
what are the negative effects on a non lti system?
For linear time invariant system, we only need to know the impulse response h(t) of the system (or equivalently frequency response H(omega)) in order to predict the output of the system in response to any input. This is done by convoluting the input with the impulse response. So a linear time invariant system is a lot easier to analyze. This is not true for nonlinear or time variant system. If the response to a unit impulse is known, the response of any discrete-time LTI system to any arbitrary excitation can be found. • Any arbitrary excitation is simply a sequence of amplitude-scaled and time-shifted DT impulses. • Therefore the response is simply a sequence of amplitude-scaled and time-shifted DT impulse responses. no more constant magnitude and linear phase and the input-output relation is conveniently described by convolution (in the time domain) or multiplication (in the frequency domain). The Fourier transform is a powerful tool for analyzing LTI systems.
the four fourier methods comparison/differences
Fourier transform - infinite signals which are aperiodic fourier series - represent periodic signals infinitly,
What is the formula for regular and inverse Fourier transform and when is it used?
Fourier transform is then used to represent a general, nonperiodic function by a continuous superposition or integral of complex exponentials. basically a fourier series except the area, deltaF, approaches 0 or letting period approach infinity
Formula to convert any time domain signal to frequency domain (DFT) (discrete time and continuous frequency) (x values will be same as the k values for the series)
N = total number of points there will be N number of X(K) - X(0), X(1)... with where the k values change and the n and x(n) change where x(n) is the y value in time e^jw is the basis function and allows for any signal to be represented as a series of sinusoids
What is QAM? explain
Quadature amplitude modultion Quadrature Amplitude Modulation, QAM utilises both amplitude and phase components to provide a form of modulation that is able to provide high levels of spectrum usage efficiency. different #QAM makes the number of bits per symbol different, such as 64 QAM using 6 bits per signal log2(64) = 6
different type of signals values and time effects
T = 1/f digital - finite amplitude values analog - infinite amplitude values continuous time - signal specified for every value of t discrete time - signal specified at discrete points on time axis
What is the coefficient for the coefficient Dn in a Fourier series?
T = 1/f (amount of time it takes signal to repeat itself)
What is the Nyquist rate?
The Nyquist rate or frequency is the minimum rate at which a finite bandwidth signal needs to be sampled to retain all of the information. For a bandwidth of span B, the Nyquist frequency is just 2 B. If a time series is sampled at regular time intervals dt, then the Nyquist rate is just assuming B is in Hz, sample at 2 times the highest frequency
what is the average value of a signal?
The X(0) or the DC component at the 0 Hz in the frequency domain
(e^-t/2)^2=
e^-t/2 * e^-t/2 =e^(-t/2-t/2) = e^-t
what is the process of find the coefficeint Dn and computing the fourier series? WHAT TO REMEMBER?
basically you need to find Dn which requires T0 plug in the f0 valyes to cancal in he basis function, integrate the function and plug in the values in the limits of integration to get your Dn you then can plug in more and more values of n to get closer and closer to an accurate representation after integrating to get Dn and you then apply the fundamental theroem of calculus you need to plug in the limits for values of T!!!! (if the periodic function is dicontinuos, you will have to break the function up into its corresponding parts)
convolution formula and range and implementing it in matlab
n = [n0 start, n1 finish] f = [f0 start, f1 finish] range = [n0+f0, n1+f1]
autocorrelation of a signal? how does it help deteremine periodicity?
correlation of a signal with itself would do a normalized autocorrelation to determine periodicity. If it is periodic with period PP you should see peaks at every PP samples in the result. A normalized result of "1" implies perfect periodicity, "0" implies no periodicity at all at that period, and values in between imply imperfect periodicity. (if the noise is of a low enough level, the output of the autocrrorelation will still look like the autocorrolation without noise) as you shifted t to minus infinity to infinity, if it matched correctly, it would equal 1 Another common method to detect the periodic signal is to use autocorrelation. This is a simple method in the time domain that you shift the signal with a time lag and calculate the correlation with the original signal (or we can simply add the two signal up to get a number, and then we can divide the largest number to scale the value to -1 to 1). If there are any periodic signal, after certain time lag, we should see the correlation have a significant increase, and a simple example is shown in the following picture. On the left, we shift the same signal to the right (shown as the red signal), and on the right, it is the autocorrelation we calculate. We notice that the autocorrelation have a downward ramp, since the number of points added together to produce the value are reduced at each successive shift. The peaks in the autocorrelation figure are showing the periodic signal and its corresponding time lag (that is how much we shift the signal to get this correlation). Let's try to create an animation next week to show how autocorrelation works )^
what is the setup in matlab to define a signal?
defines the sampling rate (number of samples a second) define the total number of seconds define the time vector(start at 0, 1/sampling rate (step), end at the last time value) define the total number of points (time*#samples second + 1 - taking into account 0) frequency vecotr( only can represent half the sampling rate (-samplingrate/2: sampl/n-1: sampling rate/2 when you amplitude modulate and you take the fourier transform the pulses are going to be the carrier frequency and the base frequency of the other signals
deterministic signal vs random signal
deterministic - physical description is known completely wether mathematically or graphically and random if some probablistic descriptions such as mean value, mean squared rather than its full mathematical or graphical descriptions
explain time scaling
dividing expands by that factor multiplying by t compresses the signal by that factor
signal energy vs signal power formulas and explanation (Eg, Ex?) (note energy signal vs regular integral)
energy - joule power - watt T = period (from -1 to 1 is 2 seconds for the period) remember, SIGNAL POWER FOR PERIODIC SIGNALS IS DIFFERENT BUT IT STILL APPLIES finite energy is energy signal (signal approaches 0) and finite power is power signal (periodic) energy signals only need to approach 0 such as e^-t/2 Eg and Ex are energies of g and x
even vs odd signals
even -> g(t) = g(-t) (flip the time and they are symmetrical odd -> g(t) = -g(-t) (flip the time and ampltiude to be symmetrical
what does a digital signal look like with noise?
even with distortion and noise accumulated over long distances the receiver can make the original signal using repeter nodes to repeat the signal allows for regeneration
period vs frequency (2pif vs 2pi/T
f = how many times a second the signal will repeat itself, where as T = 1/f how long the signal will take to repeat itself cos(2pin/100) is slower that cos(2pin/10) because it takes 1 cycle 10 seconds to repeat where the other is 100
view the spectrum of each step in double side band supressed carrier? where do the pulses end up in terms of frequency?
first one is m(t) converted to mf with the 1/2 because of the sinuoid in nature, then the 1/4 due to multiplying by carrier frequency and then multiply one more time and the 1/8 are removed by a lowpass filter there is a spectrum above+- fc and one within +- fc (USB and LSB) whcih is why there is no spectrum at the carrier frequency fc which is why it is called surpressed carrier (multiplying the carrier signal of the demodulator x 2 allows for the amplitude to be the same when modulating and demodulating) and now the pulses after the demodulation are back around the original frequency of the baseband signal
fourier transform properties to note
frequency shifting means multiplied by e^2pifjt and direc(f-f0) = e^2pijft because the direc delta in opposite domains is 1
what is the duality principle?
g(t) = G(f) G(t) = g(-f), flips domains then the time flips as well
what is the bandwidth product of two signals?
g1 has B1 bandwidth and g2 has B2, the total bandwidth is B1+B2 Hz from the convolution thereom bandwidth is the frequency difference betweeh the highest and lowest frequency in the signal spectrum
period of a signal vs aperiodic signal
here it would be 5 - starts at -infinity and continues forever aperiodic signal - signal never repeats itself
fourier transform properties table part 2
i
Explain sampling terms and in terms of graphs VIDEO
if a signal is sampled at X Hz, then X samples is taken every second frequency = cycles/sec [T] = s, so 1/T = 1/sec = frequency so if you have 10 samples a second, you have a 10 Hz sampling rate
how to convert time domain signal to frequency domain using DFT? (EXAMPLE) *how does the sampling rate relate to the discrete time vector representation?*
if you are sampling at 1 hz, then the first value of x[n] occurs at t = 0, second at t=1, etc. if you are sampling at t = 10 hz (10 samples a second) first value occurs at t=0, second at t= .1
what is the transfer function of an LTI system?
it is the response to a delta function in the time domain
How do you represent the transfer function FOR a distortionless system?
magnitude and phase this is assuming distortion less transmission when the input and outut wave have identical wave shapes within a multiplicative constnat td is a delay of attenuation y(t) = k *x(t-td) Y(f) = k X(f) e^-2piftdj Y(f) = H(f) X(f) H(f) = k ^e-2piftdj H(f) = |H(f)|e^jphi(f) so phi = -2piftd the energy of a wave is diminished over a certain length due to resistance of a wire giving off heat H(f) = |H(f)|e^jphi(f)
what is the unit impulse?
magnitude of infinity at time = 0 integral over unit impulse = 1
what is the correlation coefficeint?
max value is 1 simialrity and min is -1 dissamilarity 0 means orthogonal page 45 in textbook and integrate only over where they overlap
What is modulation? why is amplitude modulation more succeptible to noise? what is the baseband signal? what is ampltiude modulation specifically?
process moving a specific message signal into a frequency band that is dictated by the physical channel AM is more susceptible to noise because noise affects amplitude, which is where information is "stored" in an AM signal. FM is less susceptible to noise because information in an FM signal is transmitted through varying the frequency, and not the amplitude. baseband is used to designate the frequency band of the original message signal from the source or the input transducer (the transmitted signal) (baseband modulation does not use a carrier signal) ----- AM is that the information signal being carried by the carreir amplitude A(t) is linear while the phase and frequency remain constant
Eulers formula
radians, e^ix = cosx +jsinx
What is baud rate?
rate at which information is transferred in a communication channel Baud rate refers to the number of signal or symbol changes that occur per second. A symbol is one of several voltage, frequency, or phase changes.
how do sin and cos decompose into functions?
real and imaginary portion of the basis function
magnitude and phase relationship to the transfer function
rectangular = x+jy polar = r (angle) theta r = sqrt(x^2+y2) theta = tan^-1(y/x) x = rcostheta y = rsinetheta -- make sure in degree mode complex number that can represents amplitude and phase of a sinusoid rectangular = x+jy polar = r (angle) theta euler = re^jtheta
what is the cross correlation and the correlation coefficeint? how to implment in matlab?
relationship and similarity between two complex signals correlation coefficeint, p, cannot exceed 1 (1 is max similarity, -1 is maximum dissimilarity, and 0 is orthogonal) * is a convolution phi is the function of the two together basically just take the convolution of two signals, whereas auto is just the convolution with itself
How do you represent a time domain signal in a complex exponential Fourier Series? *what is a Fourier series used for?*
represent signals are which are periodic The Fourier series is used to represent a periodic function by a discrete sum of complex exponentials, if it represnets an aperiodic signal, the fourier series will repeat that signal over the selected time frame for infinity basically like the F(k) for a fourier series, for each different value of t will run through all values of n
inverse DTFT(frequency domain to time domain)
serves as a good check X(k) is the value at those points in frequency domain, n will change for each different component, (multiply by 1/N)
What are most signal composed of, and how do you determine each?
signal F = Freal+Fimaginary F* = Freal - Fimaginary cc = asterik Freal = 1/2(f+fcc) Fimag = 1/2j(f-fcc)
what is the unit step function? how does it act when multiplied? relationship to direc delta?
signal which starts at 0 notice effect of multiplying dut/dt = direct(t)
how do you simplify terms with n in them?
since e^-2inpi is usually periodic, you can find a value it always resorts to in the period and use that (the 2 pi keeps it continuously periodic) or you can use eulers formula and see the different cos and sin funcions and how they cancel out
How do you mostly plot complex functions? VIDEO
since e^-jnwo is complex = cos(nw0t)-jsin(nw0t) (real and imaginary) you can find the fourier transform to represent in the frequency domain and take the magnitude if you want, the phase angle is tan^-1(Fimaginary/Freal)
what are you actually doing when calculating the fourier series?
so you first caulacate Dn, since Dn is needed in the equaiton, then you plug in more values of n to add up over the summation and then you can start adding more values of t to be getting more accurate descriptions and each t value gets paired with all the n values etc and when you add more and more terms it is able to more closely represent a function in the time domain
What is a communication system model, and explain each block
source - originates a message (voice picture email) input transducer - converts to electrical waveform as base band signal transmitter - modifies base band signal for efficient transmission using A/D converter, encoder, modulator etc channel - medium of choice that can convey electrical signals at the transmitter output over a distance such as coaxial cable, am, FM, etc - physical waveform behaving like a filter that attenuates the signal and distorts the transmitted waveforms while attenuation increases with length of the channel noise also ocurs from unintentional interfering signals usually accomulating along the path (digital signals are much less succeptable to noise than analog signals) reciever - processes the signal received from the channel by reversing signal modifications made at the transmitter and removing distortions from the channel output transducer - converts electrical signal to its original message form
what is the period when sampling, and how is the nyquist rate applied? *Notes*
the amount of time between samples is the period of sampling, and 1/T is the frequency or can be found using the amount of time the sample occurs in one second if you sample at 10 Hz, the maximum frequency you can represent will be 5 Hz (+5 and -5 Hz on both side of the frequency spectrum)
What is pulse code modulation?
the step in sampling where you assign the number to the sample; give a code to the pulse each bit allows for 2 more patterns 4 bits = 2x2x2x2 = 16 distinct patterns or possible sample values
relationship between Frequency domain and time domain in terms of unit impulse
time scaling expansion in one domain is compression in the other domain basically means that the signal varies more rapdily in the other domain time shift in one domain is phase shift in the other
why is the carrier frequency must be much greater than the bandwidth of m(t)? what about fc>bandwidth
to avoid overlapping, if the Fc<bandwidht, there is an area of overlapping of the original signal which distorts the information as fc increases, the frequency it centers around increases causing the center of the signal to extend out into both directions making it impossible to recover the original signal
how does demodulation of a DSBSC signal work? why is it inefficent?
to recover the original signal m(t) from the modulated signal, we must retransulate the modulated spectrum back to the original posistion of the message spectrum mutliply the incoming modulated signal mtcoswct by locally generated carrier coswct followed by a lowpass filter it has a upper and lower sadband centered around Fc which takes up more bandwidth when both the sidebands are mirror copies of each other DSBSC - 1/2(cos(wc+wm)t+cos(wc-wm)t = m(t) cos wt first is USB, second is lower
explain the circuirty of amplitude modulation and the resulting graphs explain the bandwidth
we are basically trying to transmit the original signal at the carrier frequency fc of the carrier signal bandwidth is now 2B hz after the original signal had a bandiwdth of B hz, +- fc has two components (uppper side band which lies on the outside of fc and lower side band which lies on the inside) - NOTICE HOW THE MODULATING SIGNAL IS CREATING THE ENVELOPE FOR THE CARRIER SIGNAL
when is a system mutually orthogonal vs normalized? notes
when the two are equal and a constnat is involved is mutually orthognal function is normalized when kn =1 and orthonormal when kn =1 for all n
when is a system orthognal? notes
when the two are not equal (it means they are unrelated)
what is a linear time invariant system?
you can scale by constants, adding signals, and shift all at the same time or at different times and you end up with the same output
fourier transforms of demodulation and what to remember about the frequency? what about fc>bandwidth
you can use 2coswct to remove the 1/2 m(t) cos(x) = 1/2+1/2cos(x) fc>= B to avoid overlap of the modulated spectra, it must be >> than the bandwidth because you waste frequencies and overlap https://dsp.stackexchange.com/questions/2493/why-carrier-signal-cant-have-frequency-less-than-message-signal#:~:text=To%20increase%20the%20energy%20of,need%20to%20increase%20the%20frequency.&text=this%20is%20when%20the%20carrier,demodulated%20at%20the%20receiver%20end.
what happens to a signal when it is passed through a system which is not distortionelss?
|H(f)| is not constant which is a nonideality of the system (part of magnitude, phase distortion) pulse spreads out, which is known as dispersion