Chapter 5 Flashcards by L High

What is Parseval’s Theorem?

The total average power of a periodic signal is equal to it’s DC Power plus the average AC power associated with it fundamental frequency and its harmonic multiples.

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What is the AC power fraction?

The ratio of the average AC power to the total average power. AC/(DC+AC)

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What is the compact Fourier Series Representation?

f(t)=C_0+sigma[from n=1 to ∞]{C_n*cos(nw_0t+ θ_n)}

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What is the trigonometric Fourier Series Representation?

f(t)=a_0+sigma[from n=1 to ∞]{a_ncos(nw_0t)+b_nsin(nw_0t)}

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What is the exponential Fourier Series Representation?

f(t)=sigma[from n=-∞ to ∞]{D_n*e^(jnw_0t)}

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What is the name of the phenomenon where the jumps in amplitude never go away in Fourier Series representations of signals with discontinuity?

Gibbs Phenomenon

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What is the bandwidth of a signal in context of ECE3620?

The highest nonzero frequency minus the lowest nonzero positive frequency (including 0w_0 frequency if it is non-zero)

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If a signal is periodic which Fourier method is used?

Fourier Series

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If a signal is aperiodic which Fourier method is used?

Fourier Transform

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What is sinc(x)?

sin(x)/x

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What characterizes a Butterworth filter?

A minimally flat passband or stopbad.

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What characterizes a Chebyschev filter?

Rippled stopband or passband. Narrower drop off band then a Butterworth but less narrow then an elliptic.

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What characterizes an Elliptic filter?

Rippled stop and passband. Narrowest transition band out of the 3 types of filters studied in class.

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Where are the poles of a Butterworth filter located in the complex plane?

Around a circle with a radius of w_c

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Where are the poles of a Chebyschev filter located in the complex plane?

Around an ellipse

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What is the output of a sinusoidal input to a linear time invariant system?

A sinusoid with the same frequency, but a modified phase and amplitude.

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What is Euler’s identity in terms of cosine and sines?

e^(jx) = cos(x)+jsin(x)

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What is Euler’s identity in terms of cosines?

cos(x) = (e^(jx) +e^(-jx))/(2)

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What is the frequency response of an ideal delay?

|H(jw)| = 1

angle(H(jw)) = -wT where T is the delay

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What is the frequency response of an ideal differentiator?

|H(jw)| = w

angle(H(jw)) = pi/2

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What is the frequency response of an ideal integrator?

|H(jw)| = 1/w

angle(H(jw)) = -pi/2

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Why are differentiators rarely used in circuits?

They amplify the high frequency noise of circuits.

|H(jw)| = w

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What makes a bode plot special?

The frequency axis (x axis) is logarithmic, and the magnitude axis (y axis) is in dB.

What form are complex roots put in when making bode plots?

((s/w_n)^2 +2*zeta(s/w_n) +1)^(+/- 1)

What is the formula for calculating complex pole peaking at w_n in terms of zeta?

|H|db = +/-20log_10(2*zeta)

What is used to achieve single side band communication?

Hilbert transformer: -j*sgn(w)

What is frequency division multiplexing?

Signals are allocated a specific bandwidth of the frequency domain. Signals have a guard band placed between them. It allows multiple signals to travel along the same channel at the same time.

What is signal windowing?

The process of multiplying a signal in the time domain by a window to truncate the signal in the frequency domain.

What is spectral spreading?

Spectral spreading is when a signal is windowed and the signal smears inside the spectral domain.

What is spectral leakage?

When a signal is windowed and the spectral spreading smears the signal into bands where the signal should be 0.

What type of signals is the Fourier Transform used on?

Time-limited aperiodic signals

What is the forward Fourier Transform integral (from time to frequency)?

F(w) = integral(-inf, inf){f(t)•e^(-jwt)dt}

What is the backward Fourier Transform integral (from frequency to time)?

f(t) = 1/(2*pi)*integral(-inf,inf){F(w)•e^(jwt)dw}

What does the Fourier Transform inform us?

The energy/frequency density of signals.

What relationship is used to perform system analysis in the frequency domain?

Y(w) = H(w)F(w)

What theorem states that any system or signal with a non instantaneous interval in time = 0 is non-realizable?

Paley-weiner

When plotting a Fourier Series what frequencies are plotted?

Fundamental frequency and the harmonics. (Exponential Fourier Series are plotted in both positive and negative directions)

What is the symbolic representation of the frequency response used in class?

H(jw)

What is the symbolic representation of the transfer function used in class?

H(s)

What is they symbolic representation of the impulse response used in class?

h(t)

How is a distortion less system described in the time domain?

y(t) = kf(t-t_d) where k is an amplitude change and t_d is a time delay.

What is Parseval's formula?

Energy of a signal can be calculated in both the time and frequency domains. integral(-inf, inf){|f(t)|^2•dt} =1/(2pi) integral(-inf, inf){|F(w)|^2•dw}

What is the Modulation Property of Fourier Transforms?

x(t) cos(ω0t) <=> 1/2*[X(ω − ω0) + X(ω + ω0)]

What is the duality property of Fourier Transforms?

If f(t) <=> F(ω ) then F(t) <=>2*pi*f(-ω )

Given the period of periodic signal T_0. What is the signals fundamental frequency w_0?

w_0 = 2*pi/T_0

What are the formulas for finding c_n and theta_n of a Fourier series given a a_n and b_n?

c_0 = a_0 c_n = sqrt(a_n^2+b_n^2) theta_n = arctan(-b_n/a_n)

What are the formulas for a_n and b_n?

a_0 = 1/T_0*integral(over T_0){f(t)*dt} a_n = 2/T_0*integral(over T_0){f(t)*cos(n*w_0*t}dt} b_n = 2/T_0*integral(over T_0){f(t)*sin(n*w_0*t}dt}

Multiplying even and odd functions in analogous to what operation of regular numbers?

Adding even funct * even funct = even even funct * odd funct = odd odd funct * odd funct = even

How is the fundamental frequency found for f1(t) = 2 + 7 cos( 1/2 t + θ1) + √3 cos( 2/3t + θ2) + 5 cos( 7/6t)?

First determine if the frequencies are harmonically related (1/2)/(2/3) = 3/4 (1/2)/(7/6) = 3/7 (2/3)/(7/6) = 4/7 etc. These must be rational numbers. If they are rational then the signal is periodic. The fundamental frequency is GCD(1/2,2/3,7/6) = 1/6

What is the formula for finding the D_n coefficient of a Fourier Series?

D_n = 1/T_0 *integral(over T_0){f(t)*e^(-jnw_0t)dt}

How do you know if two functions f(t) and g(t) are orthogonal?

= integral(-inf, inf){f(t)•g(t)dt}=0

What is the formula of a ramp function?

r(t) = tu(t)

What is the scaling property of the delta function?

delta(at) = delta(t)/abs(a)

What is the exponential function?

f(t) = e^(st) = where s = σ + jω Using Euler's identity yields: e^(st) = e^(σt)*(cos ωt + j sin ωt)

What is the formula for the dependent voltage source caused by a mutual inductance of M in the Laplace Domain.

s*M*I(s) where I(s) is the current going through the other coil.