Circuits 3 Test 1
what is the general way to calculate convultion sums for DT? what are sum checks you can do?
- the n is the value subbing in, and positive goes right and negative goes left, opposite it was before -range of y[n] = (n0+h0, n1+h1) n0 and n1 are starts and stops of xn, and h0 and h1 are starts and stops of h[n] - n+m-1 is the number of nonzero samples, #nonzerosamples in x[n]+#nonzerosamples in y[n] remember the xn term is the whole term which isnt phase shift so convert the whole thing to m terms
energy and power of the CT unit impulse =
1
what is the magntiude of any imaginary function? how do you find the period of a Eulers formula phasor?
1 convert to time domain Acos(wt+phi) = Ae^jwt+phi
how does scaling time affect the rectangles area?
1/constant multiplied
how to solve infinite geometric series which converges
1st term/(1-ratio) ratio = nextterm/current term
DT rectangle function and # samples per rectangle
2Nw + 1
How do you determine if a system is casual? BEST WAY?
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
exponential forms of singularity functions
A*e^t-x u(t) t - negative for decay, + for growth change the rate of decay through t, and if you want to start from a constant at a certain point, add or subtract that nnumber initally adjust slope of line by A
what are the forms of continuos periodic functions?
Acos(2pift+phi) A cos (wt +phi) where w = 2pif, so you can solve for the requency and inverse, or just know that 2pi/w = T Acos(2pif/T+phi)
How do you check if a system is BIBO stable for DT and CT?
Dt = summation of h[n] CT = integral from negative infinity to infintity of |h(t)| dt, basically if the absolute of input and ouput are finite for all t, then it is stable STABLE = summable!!
L'Hopital's Rule
If the limx→c (f(x)/g(x)) is indeterminate, then the limx→c (f(x)/g(x))=limx→c(f'(x)/g'(x)) so if you plug in and get undefined, just take derivatives
Rational vs. irrational numbers
Rational = whole number or fraction Irrational = radicals, e, pi
What is the process of converting analog to digital 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 notation for a cotinunuos time function?
Standard functional notation for a continuous-time function is g( t) in which g is the function name and everything inside the parentheses (.) is called the argument of the function. The argument is written in terms of the independent variable. In the case of g( )t , t is the independent variable and the expression is the simplest possible expression in terms of t, t itself.
how to tell if a combination of continuous signals is periodic or aperiodic?
T1/T2 must be rational if it is rational, find the LCM between the two fundmanetal periods of the separate functions to find the overall fundamental period for the entire function to do this, may have to convert period to smaller units to get in the form of integers to find lcm using ti84 maintain like units
how do you deal with simultaneous translations of a graph?***!!!!
TRY TO DEAL WITH SCALING FIRST THEN SHIFT when dividing with a denominator the first way, if you scale first, just treat the shift as a constant and disregard the denominator if you have a term multiplied or divided by t only, it may be easier to shift first
*What is the sampling property?*
a time shifted direct delta times a function g(T) gives a time shift to if there is no shift, it is just g(0)
arithemtic of even and odd functions
add and subtract- checl ratop pf [eriods
how do you move a graph up or down?
add or subtract a constnt to the end, + goes up and subtracting goes down
associatative property cumulative sum DT or CT what do these properties allow?
allow for the output of any cascaded system to be y[n] = x[n] con (h1 con h2)
explain how negative values affect ampltiude and time scaling as well as time shifting
amplitude - flips over x axis scaling - reverse across y axis (g1 goes t g-1, g2 goes to g-2) shifitng - minus moves origin to the right, positve moves origin to the left
What is the use of the unit impulse function? How does it act?
area is also referred to as the strength impulse with a strength of one is th eunit impulse
what is the strength of the direc delta?
area under the curve
what are the three ways to handle 2nd order DE w/ constnat coefficents for gneral solution? also how to do for DT?
ay''+by'+c=0/ar^2+br+c=0 convert ys to r y''-3y'+y = r^2+r+constant use quadratic formula for to solve for the two roots x = (-b+-sqrt(b^2-4ac))/2a or factor case 1: two distinct real roots r1!=r2 yt= c1e^rt+c2e^rt case 2: complex conjugate roots r1,2 = y+-ix yt= e^yt[c1cosxt+c2sinxt] case 3: repreated roots r1 = r2 yt = c1e^rt+c2te^rt time shifted values for discrete are the lower squared values
how to tell if a DE is linear? bad
ay1''+by'+c = x(t) dont have any functions of y (siny) or dependent variable dy/dx (x is indepedent) y is one differentaiting, therefore dependent no y^2 terms or y' * y ys can be differnt form of differentiation
BIBO check #2
basically let x(t) = unit step if you get a unit step output, it is bounded sunce the unit step is bounded
how does time scaling (compression) work in DT?
basically losing samples
what are the periods of constant functions, such as the sgn, unit step, rect, etc?
constant functions ARE periodic but there is no period, so when they are under an integral, they = 0
when is a combination of continuos time signals periodic? When is a discrete time signal periodic, as well as a combination? what is the period of each?
continuos combined are periodic when the ratio of the periods are rational once you know this, the period is the LCM of the two periods discrete time are periodic when the frequency values are rational (after factoring out the 2pi) once you find the frequency, if in an m/p form, p is the period discrete combinations can be the same if you can add an integer, m, to get the second frequency, such as F1+m =F2
What are the two types of signals? How do you tell from a graph when either is being used?(notation, name, examples, etc.)
continuous time variant signal - x(t), amplitude changes continously for each independent axis increment with time ranging from negative to positive infinity, like a line graph, defined at every instant over a time interval discrete time signal - x[n], deals with integers, could be used for looking at temperature once every few seconds, stem plot look at the axis and see if it is using n or t
what are the types of values a signal can take? Where are they found from? How do you tell on a graph which is being used?
continuous value - amplitde takes any value discrete - amplitude can only take certain values, such as a 0 or 1 find from the y axis if the y axis value or constalty at a flat line and then either drop or rise to another flat, they are discrete whereas if they are curvy, it is continuos
*what defines the unit impulse in CT?*
defined by strength (area under the curve)
how to scale the time in a graph? (review) how do you think of it
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
how to find even and odd components of a signal graphically?
draw the g(-t) and -g(-t) version and combine them together for each formula easier: list all g(t) terms with corresponding t, flip all values for g(-t) and plug into the forumla for new g(t) terms and plot this new graph for -(G-(t)) dont do double negative and make positve because -t only applies to the inside
phase shift of even and odd signals
even - none odd - pie >0 -pie<0
what is the even function vs the odd function? What to remember when doing this?
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 if one side of the equation is 0, the counterpart is the function type you can factor out the negative from the t!!! but you can also put negative in parnetheis with the (-t)^2 like if it is squared to cancel it!!!!! and only apply to the t itself when for the t!!! apply negative to whole thing for the odd
what things are the same for DT vs CT?
even and odd, periodic signals can also be in discrete combinations of even and odd amp scaling time scaling
general convolution method example
example continued
how does time scaling expansion work in DT?
expand each term by factor divided by, and between the integers are 0
how does time scaling work when it is a ratio of integers, such as 3x/5
expand first by 5, then compress by 3
*what is the scaling property of the impulse?* REVIEW what to remember about unit impulse and periodic unit impulse?
factor out constant multiplied by t everytime before graphing, will affect amplitude instead of shift, and then you can integrate etc
easy way to check for time invariance?
first apply system response then shift to xt, and then reverse order and htey should be equal for time invraiant, remember you only shift the xts when going through the system after it has been shifted
fundamental period, frequency, and angular frequency in F0 vs fs
fo, To, w0 - cycles second fs = samples second
forward differnce vs backwards difference of DT functions
forward = next sample - current backward= current sample - previos go g3-g2 would be for gt, etc
How do you scale the amplitude? (review)
g(t) * A is when you know its the amplitude becasue it is the entire function every g(T) value x A - flips, it is multiplied by every point
what is the unit rectangle? (gate functions)
height, widht, area are all 1
explain amplifier!!!!!, summing block, and integrator
highest order may tell you how many integrators you need amplifier can affect outputs of summing blocks, dont change yn values usually as well
what is linearity?
homogenous and additive
what is a time invariant system?
if a system is initally in its zero state and an arbitrary input signal xt causes a response yt and in input signal x(t-t0) causes a response y(t-t0) for any arbitrary t0, the system is to be time invariant shifting doesnt change amplitude
what is the delay block? WHAT TO NOTICE?
if the arrow is going backwards, it is +!
what to remember about time scaling?
if the whole t term in the g(t) has a denominator, not sure when to disregard? only disregard if you scale first then shift, otherwise, if you shift first treat it like it is under btoh terms also does not have to get smaller on both sides, rather just shifts to one side, if a point is originall at 1 and you divide it by 4, it moves to 1/4, where as point 4 moves to 1 if it is divided by a number <1, then it will be stretched, such as being divided by (1/2). g1 moves to g2 etc.
how do you shift graphs horizontally??
if to>0, t is positive and shifts to the right to (t-to) if to<0, t is negative and shifts to the left to (t+to) shift origin to to, everything else stays the same just place the origin and follow from there
what is important about the unit step response?
if you have the unit step response, then you are able to plug in values of the unit step into the unit step response transfer function turn rectangle funcitons to unit steps
energy of time limitless signal
infinte
if an amplifier term applies to a whole line, what do you do?
inverse it so you basically factor it out
convolution integral continued WHAT?
just basically find the area where they voverlap for each term!
difference between x(t/2+t0 and x(t+t0/2)
just write it as x(x+t0)/2 using common denominators and do the amplitude scale, then time scale, then time shift method
How do you go from DEs to block diagram?
label all branches from the input and output side and try to work backwards, and the last adder usually combines the last two solve for the highest order output signal, or just output signal itself
How to plot in freuqncy domain for phase and magnitude? (complex signals)
magnitude, treat as regular for rectangle funciton if f is multiplied, make sure to divide each term by what it is multiplied by phase phase slope is what is with the e
what are the forms of discrete time sinusoids and what is the fundamental period? WHAT TO REMEMBER?
make sure to factor out the 2pi for the period portion just if it is conitnuos and periodic does not mean it can be discrete and periodic *A discrete time sinsoid may not be periodic, only periodic when k is a integer or ratio of integers, with the bottom number the period*
How to test for homogenous equaiton? bad
multiply x values by k, if it comes out to solution of y*k, it is homogenous
what is a homogenous system?
multiplying the input signal by any constant multiplies the zero state response by the same constant
is an incremental system linear?
no, but it can thought of as a linear system with an offsert
Convultion in CT steps and tips
nonzero values to test are (n0+hn0, n1 + hn1) original h values remember you are actually multiplying the overlapping areas, and finding the actual area + numbers for t shift right and negative shifts left it only works for where they overlap, not the overlapping of lines but actual area, just multipy the amplitudes and then find the area
what things are the differnt for DT vs CT?
not all sinusoids are periodic in DT sinusoids can have different period with same DT graph unit impulse time scaling(expansion and compression) differencing (derivatives) and accumulation (integration)
difference betweeen directdelta[n] vs directdelta[an]
nothing, time scaling doesnt affect it
casual/noncasual combinations*
only present or past values for casual present and future for noncasual if you get only present for t=0, shift to t=1, then determine again and keep intcremening t until you get a combination if it is only presnet for infinity, then it is casual all three types, infinty basically you can choose any value of t, just make sure the output and input both only depend on the same time or lower time values
what does the minus sign apply to in even and odd formulas?
only the t, but can be removed if there is a t^2
other way to check for homogenous equation easier
pass the input x1 into the system h, observe the output as y1, multiply y1*k pass the same input in the system with a constant, if it does not turn at as ky1t, it is not homogenous basically pass then multiply, and then multiply then pass and should come out the same
other way to check for additivity
pass two separate inputs through the system and add them up, this is the result you should get when they are passed through togther, otherwise it is not additive pass 1 input, x1 through the system h and observe as y1t pass 2nd, x2 input through system h and observe as y2t if additive, y3t = y1t + y2t therefor, pass x1t + x2t through the system , if it is not outputted as y1t + y2t, not addditive
What is the Signal power for DT and CT? what signals is it used for?
periodic time infinte, just find the period and pick any two points in that period
how to tell if polinomials or exponentials are periodic?
plot, and if they repeates themselves, they are periodic
explain causal systems
plug an input in, that is the present, any values appear on ouput which are greater than the input, it is noncasual
how to check if systems are casual vs noncasual
plug in the present time shift- all xs or ys should only depend on the present or past periods, can be ahead any of the phase shifts basically, plug in the last present time value, or the highest t-t0 value or after the time it is excited solve for the lowest order output usually only present or past values for casual present and future for noncasual if you get only present for t=0, shift to t=1, then determine again and keep intcremening t until you get a combination if it is only presnet for infinity, then it is casual
how does the unit rectangle act when multiplied?
product is zero outside its nonzero range and is equal to the other function inside its nonzero range
convultion of function* direc detla DT and continous what ways?
remember if it is factored out with a convolution it turns into a direc delta
how to go from graph to a function that has been time scaled?
remember that when you divide t/constnat, it moves each point a factor of that constant, 1constant, 2xconstant, 3x constant when you multiply, it condenses, so if it shrunk, multiply by that factor
how do you find if a system is bounded given a step response? how do you find the response to a certain input?
remember the chain rule remember if the g(t) * direc delta and g(0) = 0, then the whole thing equals 0 becuase the direct delta only has a vaue at find the unit impulse repsonse by taking the derivative, once you have the unit impulse reponse, you can take the conolution of input response and unit impulse response to find the output or if you have a step response output and another step resposne input, you can simply plug it in
how do you do convolution which increases to infinty?
remember to section integrals off from the overlapping
how does amplitude and time shifting work in DT?
same as CT
sampling property vs convoltion property
samplying is integral of direct(t-t0) xt = x(t0) x(t)*direct(t-t0) = x(t-t0)
how do you use the sum by column convolution method when it is periodic?
so if the h[n] has x number of terms, dont start adding for the period until it has 5 number of terms and only continue for the T terms, depending on the period of the x[n] if the order is 2 1 3 2 1 3, T = 3 if the h[n] only has 2 columns of the right amount of numbers but needs another to satisfy the amount of periods, repeat the period again
explain invertiblity of signals
solve for the input if you can get the input from the output, it is invertible if you can accuratley represent input response for all values of t, (solve for x[n]), it is invertible doesnt matter if x[n] is shifted
explain the memory of signals
static only when the output depends on the same input time dynamic when output depends on the same input time and other times as well x(t) = y(t) static since y(0) = x(0) dynamic if x(t) = y(t-1) x(0) = y(-1) derivatives do have memory, same with integrals
What is the correct way to check if something is BIBO stable?
summable only! or integral of Ht of the impulse response!!!
how to remember whcih way something shifts?
t-t0 basically says shift origin to t0, set equal t0 - Tao, when you set these two equal, it is backwards
how to plot magnitude and phase of a complex signal or Real function?
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)
what defines the unit impulse in DT?
the amplitude
what is accumulation in DT signals?
therefor, unit step is analogous to antidervitives, being that the accumulation of impulse is the step just add up all terms, can decrease summation from -infinty to n of h[m] = g[n]
what to remember about singularity functions before 0?
they extend infinity in both directions unless stated otherwise
what important to remember about singal energy and power?
they have magnitudes so if given discrete points, only take the positive values of them or use absolute values when plotting
What is the signal energy for DT and CT? What signals is it used for?
time limited dont square inside part, just the outside variable
use energy of une function to find dependence on the other
undo what was done to it from from inside out, but time shifitng does not affect remember that negatives go inside parenthesis
What is the combination unit periodic impulse definition?? REVIEW
uniforml spaced infinte sequence of unit impulses Any time ther is a subscript, it is periodic impulse with the period being the subscript space them out according to there nT period when constants are factored out using the scaling property, period is also divided by the constant
*how does the integration of t extend for each derivation?*
unit singlet is the direc delta unit doublet, u1t, is the unit step unit triplet, u2t, is the unit ramp
how are complex exponentials and sinuosids related? what do when you need to use them or integrate?
use eulers formula to convert
check for stable vs unstable thorough explanation
use ut as input, plug in, if it doesnt turn out as bounded output such as ramp, it is unstable
unit impulse response
using the impulse function as the h(t)
review phasors
v(t) = Vm cos(wt+phi) = V~ = Vm <theta= Vm e^jtheta at some frequency if a j is in the amplitude, add 90 degrees to the angle if it is a negative j, subtract 90 degrees from the angle rectangular = x+jy polar = r (angle) theta euler = re^jtheta rectangular = x+jypolar = r (angle) thetar = sqrt(x^2+y2)theta = tan^-1(y/x)x = rcosthetay = rsinetheta -- make sure in degree mode THIS HOLDS FOR FUNCTIONS OF COMPLEX NUMBERS SO YOU CAN PLOT THEM
what do you need to remember about time shifiting?
when you shift, it centers the origin at the number, so the rectangle is centered there as wll as the triangle etc.
how do you convert to the s-domain for a transfer function? what is the transfer function itself?
x(s) h(s) = y(s) set h(s) = y(s)/x(s), and set that for the s values as well
when may two discrete time sinusoids be the same? what does this mean? how does this affect the graph???? *REVIEW*
you can add any *integer* multiple of 2pi and the sinusoid is unchanged in continuos, the graphs look different, but if you use the table on the calculator you just match you can factor out the 2pi and solve that the periods are within integers of each other they bascially will be the same when there is an integer x 2pi separating them, so just factor out the 2 pis to check on the periods
how does an even or odd function become affected when integrating for area?
you can do 2*integration from 0 to upper limit odd, the integral over the whole thing
what is the sum by column convolution method? when do you use it?
you know amplitude and shift, convert it for the h[n] with the proper shift, and use the h[n] values in the new found h[n] function and then add up if you have 0s in either row, shift it over that much on the graph, 1 0, shift over to the left once etc or just write plus for it' or you can just find the nonzero range and apply it to the values
What to remember about periodic impulse when using scaling property?
you still affect the period every time you factor out an constant even though the period is not specificed in the problem, if there is a plus or minus a value, you shift it to the value still
how do you deal with complex exponetials? what is the form for discrete? what is the general form?
you use eulers theroem; e^jwt = coswt + jswin wt plot the real component and imaginary of each can also be Az^n, where z can be things such as e, etc.
