Circuits 3 - Test 2
what does the harmonic number k start at for a fourier series?
0!, if period is 4, it is 0 1 2 3
what is double sideband transmitted carrier modulation
1 could be a k value As mentioned in the previous section, double-sideband suppressed-carrier modulation is not widely used. A modulation technique that is widely used is double-sideband transmitted-carrier (DSBTC) modulation. This is the technique used by commercial AM radio transmitters and by most international shortwave transmitters. It is very similar to DSBSC, the only difference being multiplication of the modulation by a factor m and the addition of a constant K to the signal x(t) before modulation (Figure 12.8).
what is the half power mark on a boding daigramn?
3dB 20log(1/sqrt(2)) = 3 or -3db
when is a system causal?
A system is causal if the output does not anticipate future values of the input, i.e., if the output at any time depends only on values of the input up to that time. ouput cannot occur at any times before, if you plug in 0 and only get present times, plug in more until you get combinations if you plug in 1, it is present just view the input signal, if the output signal tries to occur before input signal, it is not caual if there is no x+1 or y+1, then it is probably causal
fourier transform of a constant
A*direct(t)
convolution vs Fourier Series
Acos(2pit /T0 + phi) or they can be complex sinusoids of the form Ae j2pit /T
practical lowpass and bandpass passive filters and meaning
All the practical fi lters we have examined so far have been passive fi lters. The term passive means they contained no devices with the capability of producing an output signal with more actual power (not signal power) than the input signal.
Cr is ___ and RC is ____
CR is rdifferentiator (highPass) RC is integrator (lowpass)
difference between continuous and DT Fourier series *what to note?*
DT is over a finite value K (period) whereas continuos is -infinity to infinity using eulers, you know that this will be equal to 1 (cospin) =1 and sin (pin) = 0 where n is any integer k values = Nf in length in CT, it is an infintie value of x[k]s, but x[k]s in Dt will just repeat so there is no need to sum them all in DT say you have a period of 4, then x[0] = x[3] x[1] = x[5], x[2]=x[6] x[3] = x[7] etc, so you dont need to factor in those k values outside of the period (it starts from 0)
DT period and frequncy
F0 = 1/N0 (period)
difference between four different fourier methods
FS = integer k component sampled in harmonic number , periodic DTFS = DFT basically FT = aperiodic, true for all f values from 0 to 1 period approaches infinity
*what is an ideal filter?*
Filters separate what is desired from what is not desired • In the signals and systems context a filter separates signals in one frequency range from signals in another frequency range • An ideal filter passes all signal power in its passband without distortion and completely blocks signal power outside its passband all ideal filters are noncausal!!! because the impulse response is applied an infinite time before 0 (unit sinc magnitude is constant and phase is linear!!
conept of Fourier Series vs Fourier Transform
Fourier Series is used for periodic signal to represent infinitley, whereas Fourier Transform is used for aperiodic signals to represent infinitly
what is angle modulation?
In angle modulation, instead of the information signal controlling the amplitude of the carrier, it controls the phase angle of the carrier.
phase delay vs group delay
In signal processing, group delay is the time delay of the amplitude envelopes of the various sinusoidal components of a signal through a device under test, and is a function of frequency for each component. Phase delay, in contrast, is the time delay of the phase as opposed to the time delay of the amplitude envelope.
2 forms of DT fourier Series
N = No*m where m is any integer used to represent periodic DT signals
2 different forms of Continuos time fourier transform and the differences
NOTE: Both form graphs are vs F
process of converting analog to digitial signals
Sampling - use and record sample of data at certain incremented times Quantitizing - for each continuous value, round to the nearest discrete value for each sample Encoding - once sampling is done to get the discrete time and quantitizing is used to get discrete values, you are able to encode data to a certain binary sequence
*What is the Continuos time fourier series defintion? What is the notation?*
T is the fundamental period if the function is periodic, if you choose the fundamental period as the time window, then the FS representation of the function can be represented infinity the time window chosen for aperiodic is the the repeated signal in the FS
What is the uncertainty principles?
The delta functions are "localized" in time; they are nonzero at just one point and zero everywhere else. But the frequency "spread" of the delta functions is not localized. We showed that X(ω) is always 1; it never dies out. For sinusoids, the opposite is true. They never die out in time, but the frequency spread is just one point. The pulse function was somewhat localized in time, and somewhat localized in frequency (the sinc function dies out asymptotically). g This is the Heisenberg uncertainty principle: the product of the time "spread" and frequency "spread" of a function can never be less than a defined minimum nonzero value
Why is the delta function important?
The delta functions contain all frequencies at equal amplitudes. Roughly speaking, that's why the system response to an impulse input is important: it tests the system at all frequencies.
Definition of the Continuous time fourier Transform process of finding fourier transform
The salient difference between a periodic signal and an aperiodic signal is that a periodic signal repeats in a finite time T called the period. It has been repeating with that period forever and will continue to repeat with that period forever. An aperiodic signal does not have a finite period. An aperiodic signal may repeat a pattern many times within some finite time, but not over all time. The transition between the Fourier series and the Fourier transform is accomplished by finding the form of the Fourier series for a periodic signal and then letting the period approach infinity. Mathematically, saying that a function is aperiodic and saying that a function has an infinite period are saying the same thing.
What is double side banded suppressed carrier de modulation?
This demodulation operation is a good example of the advantage of using transform methods that include negative frequencies. In this case some of the spectral peaks are shifted from negative to positive frequencies and vice versa and directly indicate the correct demodulated signal
Concept of the Fourier Series
We can improve the approximation further by adding a sinusoid at a frequency of twice the fundamental frequency of x(t) (Figure 6.4). If we keep adding properly chosen sinusoids at higher integer multiples of the fundamental frequency of x(t), we can make the approximation better and better and, in the limit as the number of sinusoids approaches infinity, the approximation becomes exact (Figure 6.5 and Figure 6.6). the next would be 3f, and then 4f, all the way to infinity, so increase multiples of f or decrease for multiples of T *basically, you have a periodic function which can be represented by a summation of sinusoids at different frequencies*
what is an active filter? examples
With the use of active devices the actual output signal power can be greater than the actual input signal power
important property on conjugation in DFT
X*[-k] X[k] = X[N-k] because it is periodic in DT and it repeats so find a negative value, and the conjugate of it should give you your asnwer
find fourier transform from fourier series
Xp[k] =fp X(kfp)
what is single sideband suprresed carrier modulation?
Y( f ) = (1/2)[X( f − fc ) + X( f + fc )]H( f )
output signal of double sideband transmitted carrier modulation
Y( f ) = (KAc /2){[delta( f − fc ) + delta( f + fc )]+ m[X( f − fc ) + X( f + fc )]}
What is double side-banded suppressed carrier modulation? how does it work? what is frequency multiplexing?
YDSBSC ( f ) = (1/2)[X( f − fc ) + X( f + fc )].
How does time scaling work in a Fourier Series Case 2?
a>1 (compressed-loss of data) 0<a<1 (expanded-add 0's) same as before
absolute vs halfpower vs null bandwidth
absolute - the largest frequency component minus the smallest frequency component null - The difference between the frequency of the first null (zero magnitude) in the frequency spectrum above and the frequency of the first null below is called first null bandwidth. half - The half-power point or half-power bandwidth is the point at which the output power has dropped to half of its peak value
multiplication duality CTFT and importance
allows difficult to find convolutions to be found using Fourier Transform
DC component using Fourier TRansform or area under a curve (average value) ways to do it?
also known as the total area basically just if you are looking for a total area for a hard integral, you can find the Fourier trasnform and evaluate it at F or w= 0 The average value of any periodic signal is the value of its CTFS harmonic function at k 0. 16. In Figure E-0 is a graph of one fundamental period of a periodic functio x t A CTFS harmonic function cx k is found based on the representation time being the same as the fundamental periodT0 can be applied to all different forms *look physically at the graph at the 0 value of t*
lowpass, highpass, and bandpass filters frequency responses and trasnfer function *REVIEW TFs of each review graphs* REVIEW STUPID ****
bandpass = cascading highpass and lowpass filter highpass is top (1-lowpass = highpass) lowpass is bottom bandstop is parallel lowpass and highpass page 516 has definitions
ideal bandpass, bandstop, lowpass, and highpass filter transfer functions with phase and magnitude explanations
bandpass has a pass band in midrange of frequencies and stopbands at low and high bandstop reverses that pass and stops of the bandpass passbands- filter allows signal power to pass unaffected, where stop band attenuates the signal power, allowing very little to pas through magnitude is constant and phase is linear
covnovlution of shifted direc deltas *REVIEW*
basically just add or subtract the time delays and almost as if you are foiling the signs out
CTFT and DTFT relationship
basically just sampling
what is the transfer function in a cascaded system?
basically you are able to multiply terms instead of convolution them due to convolution-multiplication duality property
time integration on a fourier series 2 cases
case 1 = DC component =0 (still periodic) case 2 DC component != 0 (aperiodic)
bode plot axis example
constant, just a straight line
how to do block diagrams to figure out transfer functions?
convert everything to jw
no need to scale into account for sin and cos review why multiplying in time domain means what
convolution in frequency domain
what is the power ratio of transfer functions? units?
dB
fourier transforms of periodic signals what does it consist of? *REVIEW*!!
direct delta impulses, it is summation of impulses!
f form vs w form for foureir transform, REVIEW!
divide by 2pi you are in w form
time scaling from test 1
dividing by a number stretches by a factor of that number, if x1= g(t) is scaled by x2= g(t/2) for t=1, whatever happened at t=1 will happen at t=2 for x2 since that will give both g1 each point multiply by 2, g1 will move to g2, g2 will move to g4 etc if g(t/x) where x = 2 shifiting amplitude to those points if it is <1, it multiplies, ultimatley compressing by a factor of that number basicalal if g(2t), say that g(t) = g(t*2) so at t =8 for the first will need to happen at g(4) for the second each point will be divide by 2, so g2 will go to g1, g8 will go to g4 if g(t/x) where x = 2 notice it as acting on the t itself so say t is divided by 2, whatever was originall at g(1), is now at g(1/2), g(2), is now at g1, g(4) is at g(2) if you multiply by a negative the time is reverse and flip the curve horizontally negative flips across
Euler's Formula what?!!?
do not include js because j is already taken into account
ROC for delta functions
everywhere but the origin
output of rc circuit as lowpass filter in time domain
explain in book on page 519
frequency and wavelength relationship
f=lamda/c
fourier transform of cos and sin
factor out the 1/2 of j/2 and then multiply by the leftovers to get into the correct form anytime they are phase shifted use these
how do you represent aperiodic DT signals, and what is the process?
find the DTFS, and let the period approach infintiy and that is the DTFT basically the same except it is over infinte time period for a DTFT
are ideal lowpass and highpass filters causal or stable? how to determine with example?
find the impulse response the h(T) is a unit sinc which is the direct delta applied at 0, the unit sinc is infinite, so it responds before 0 so it is noncausal unit sinc is absolutely integrable (1 - area under the curve) so it is stable • Ideal filters cannot be physically realized, but they can be closely approximated. rect(t) is not causal whereas rect(t-1/2) is basically just apply fourier inverse transform and if it does not occur before 0, then they are causal
How does chaning the period affect a fourier Series?
first signal defined by 0f, fo, 2fo, 3fo... second signal is defined by 0f, 2f0, 4f0 with 0s between remember the 1/2T0 works becuase you are dividing the frequency and expanding the period
ideal filter defintions
fm is the frequency marker highpass
What is the average power in a DFT vs DTFT?
for DTFT or any Fourier Transform, the average value of x[n] is X[0] in the frequency domain since DFT has different placement of N, it is X[0]/N
since the Fourier Series is periodic, how do you represent Aperiodic signals?
fourier transform can represent any signal for all time as long as x(t) is bounded
fourier transform of the comb function
fourire transform of dirac is a constant, you have a shift, so you basically just shift the other values values being convolution using the shift
What is linearity?
homogenous and additive(constant multiplied as well as adding two inputs also adds two outputs)
fourier transform of truncated pulse
http://www.thefouriertransform.com/pairs/truncatedCosine.php
What is important about the period of a Fourier Series?
if a signal is periodic and the time window you choose is its period, the Fourier series will repeat it infinitly since the Fourier Series is periodic itself, so make sure to choose the correct fundamental period if the signal is aperiodic, the FS will repeat the signal, however, it will only match for the indicated time fame
bode plot numerator and denominator slopes of magnitude and phases review phase of all, jw!! how to plot laplace transform?
if it is squared, it is 2*20, or 40 log(|H(jw)| same for phase, slope doubles 0 until it reaches the number it is over for phase, it is tan^-1(w/x) where the is it s factored to (1+jw/x) axis may have to be adjusted considering where is starts, convert all s to jw
How does time scaling work?
if you compress in one domain, you expand in the other and vice versa
When are two functions orthoganol?
inner product =integral of complex conjugate
What is an LTI eigenfunction and eigenvalue?
input an arbitrary function, e^st, you get the same function multiplied by a constant
how to get unit impulse response from frequency response?
inverse FT of its frequency response, and you can plot them to compare
What is the discrete Time fourier series have different in terms of multiplication convolution duality?
it deals with circular convolution which means instead of a convolution being over infinity, it is just over 1 period
What is the energy or power of a Fouerier Series? or any Fourier Series, transform Etc. what to note?
it is a power signal since it is periodic // Parvesals = Power theroem The average signal power of a signal is the sum of the squares of the magnitudes of the harmonic function values. notice you take the magnitude of the function!!
How do you find the transfer function?
it is the fourier transform of the unit impusle response h(t)
physical process of finding Fourier Series part 1
just increase the k value
unit sinc defintion tri(1/2) or any function at a point *MULTIPLY BY PI TO MAKE IT THE UNIT SINC
just plug in that value of t on the graph and the amplitude is the answer you can plug into calcauator like that to graph
how does time scaling work in a fourier Series case 1?
just think of as before Acos(t/a) is really decreasing frequency so increasing the period now we are decreasing the period which is increasing the frequecny, which is like Acos(at) for the particular exmaple
fourier inverse of jw or 1/jw
jw = derivative 1/jw = antiderivative
what is the transfer function for a system in CT?
k is the order of derivative
how to find average singal value of FS or signal with k in it?
let k =0, then basically just cancels out
how does time shifting occur in a fourier series?
magnitude is the same, where phase is shifted from phase values giving different slopes
causality and ideal relationship
noncausal means ideal causal means nonideal
time shifting results in what for the frequency domain? how ti frequency shift?
phase shift
*write a rectangle given a graph*
rect((t-middle)/period)time it took
unit rectangle and sacaling remember not to square the area if doing parsevals function of the rectangel, just square ampolitude and then multiply by the area
rect(t/2) goes to -1 to 1 rect(2t) goes to -1/4 to 1/4
what to note about DTFS?
remember to actually do the reciprocal if you are using the period or go back and forth between the two
complex exponential response in CT?
s = sigma + jw (real and imaginary compoents)
FT time shifting REVIEW
same as Fourier series, multiplied by complex which only affects the phase positive is the adding a t value, and it can be e^jphi with any function
what is the number of harmonics needs for a DT signal?
same as its fundamental period for DTFS method
how do you plot mangitude and phase of a graph? REVIEW how does it work for direc delta?
take magntiude first (plot using calculator for sinusoids) remember magnitude is positive only since it is the absolute value, so if it goes into the negative, flip into the positive! actually take magnitude of some numbers and only the real portion since the magnitude of complex exponential is 1 from eulers formula you can just do absolute value in calculator of the function in for you magnitude phase slope is whatever the e is raised to, but also the other funciton cant be 0 and keep minus signs with it, because eulers form Ae^jwphi so between 0s of the magntiude graph is where the phase repeates itself, and it only has values where the magnitude is not 0!! remember to keep the negative signs in the exponent!!! only the imaginary portion (no imaginary means no phase) if it is only real and even, the phase will be odd, and where it is positive it is pie, and may have to flip for odd signs if in form of a+bi, tan^-(b/a) *if e^jwt is posititve, it is a positive slope, negative is a negative slope for the phase, and is basically e^jphi for direc deltas, the magnitude is easy but the phase is dots of the amplitude where the values occur
difference between DTFS and DFT X[#] means what?
the k value at the DFT or CFT is equal to the # note that difference, and if you are given DFT solve for DFT by dividing the N over anytime you see x[k] used, just plug in x[k]/N
prove the convolution property of continuos fourier transform
the last part comes form the definition of the fourier transform
How do discontinuos signals act in a fourier series?
there will always be a tiny ripple near the discontinuities of the original signal
boding plot example what to remember about cascaded systems?
this allow us to add all the multiplied terms to allow for a summation
what are you plotting on a boding diagram?
transfer function always factor to form of 1+jw/10 x axis = log w, y axis = 20 log|H| for magnitude both w are in increments of (10) = .1, 1, 10 , 100... type deals y axis is in terms of 10, 20 etc. in dB.
*what defines a distortionless system?*
uniform frequency response and linear phase Therefore a distortionless system has a frequency response magnitude that is constant with frequency and a phase that is linear with frequency (Figure 11.59). The magnitude frequency response of a distortionless system is constant and the phase frequency response is linear over the range − < < and repeats periodically outside that range. Since n0 is an integer, the magnitude and phase of a distortionless fi lter are guaranteed to repeat every time changes by 2.
-How is the sinc function derived? *sinc(0)?*
unit sinc becuase height and area are both 1 1 (using lhopitals rule) take lim t approaches 0
What is the form of the answer to a Fourier Series? what is the easiest way to represent it?
usually complex, so represent it by its magnitude and phase in the frequency domain
Trigonemtric CTFS
where ax[0] is the average value of the signal in the representation time, k is the h armonic number and ax[k] and bx[k] are functions of k called harmonic functions.
explain the last property, both parts on the table
x(t) e^2pif0t is a shifted x(f)
impulse sampling formula
xdelta(t) = x(t)delta ofTs (t).
does x[k] = x*[-k]?
yes
2 transfer function forms
zeros - numerator makes 0 poles - 0 in denominator makes infinity
how do you represent a bandlimited periodic signal?
• If a bandlimited periodic signal is sampled above the Nyquist rate over exactly one fundamental period, that set of numbers is sufficient to completely describe it • If the sampling continued, these same samples would be repeated in every fundamental period • So the number of numbers needed to completely describe the signal is finite in both the time and frequency domains
