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40 Cards in this Set
- Front
- Back
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Signal |
a representation of information. A function of time (usually) or space |
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Sets of numbers |
R: set of all real numbers (a number whose square is non-negative) Z: set of all integers {...-3, -2, ,-1, 0, 1, 2, 3...} N: set of all natural numbers {1, 2, 3...} W: set of all whole numbers {0, 1, 2, 3...} Q: set of all rational numbers {m/n such that m, n are integers and n is non-zero} Q': set of all numbers whose square is positive but cannot be represented m/n. Notice Q + Q' = R. |
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Continuous time signal |
the domain is the set of all real numbers for a given interval of time |
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Discrete time signal |
the domain contains a "countable" set of values |
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Reflection of x(t) |
x(-t), does not change the area under the graph. integral of x(t) = integral of x(-t) |
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Even functions: What are they? Is their linear combination even or odd? Is their product even or odd? What is the shortcut for the integral? |
x(-t) = x(t) A linear combination of even functions is even. The product of even functions is even. The integral of an even function is twice the area under the positive (or negative) portion. |
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Odd functions: What are they? Is their linear combination even or odd? Is their product even or odd? What is the shortcut for the integral? |
x(-t) = -x(t) A linear combination of odd functions is odd. The product of odd functions is even. The integral of an odd function is zero. |
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Formulas for even and odd functions: Splitting an equation into even and odd parts. Adding and multiplying even and odd functions. |
x_even(t) = (1/2)[x(t) + x(-t)] x_odd(t) = (1/2)[x(t) - x(-t)] The sum of an even and odd function is neither even nor odd. The product of an even and odd function is odd. |
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Time shifting |
x(t) --> x(t-b) Graph shifts to the right b units. |
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Time scaling |
x(t) --> x(at) Graph shrinks by a factor of a (each horizontal component divided by a). |
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lowerbound |
the lowest value in the domain of a function. If x(t) >= M for all t, M is the lowerbound. |
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upperbound |
the greatest value in the domain of a function. If x(t) =< M for all t, M is the upperbound. |
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Unbounded signals |
signals whose upper or lowerbounds are infinite |
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Bounded signals |
signals whose upper and lowerbounds are finite |
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Formula for power of x(t) |
[x(t)]^2 |
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Formula for energy of x(t) What are signals with finite energy values called? |
integral of [x(t)]^2 dt Signals with finite energy values are energy signals. |
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Formula: Average power of periodic signal x(t) |
(1/T) integral over one period of [x(t)]^2 dt |
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Formula: Average power of nonperiod signal x(t) |
lim as T--> infinity of (1/T) integral from -(T/2) to (T/2) of [x(t)]^2 dt |
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Formula for L^(p) norm. What is L^2 norm? What are signals with finite L^2 norm values called? |
[integral x^p(t) dt]^(1/p) L^2 norm is power. Signals with finite L^2 norm values (power values) are power signals. |
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What should x(t)^2 be replaced with when calculating values for complex signals? |
|x(t)|^2 |
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Deterministic signal |
a signal for which the signal value can be determined for a given value of time (i.e. signals with mathematical expressions). Can be evaluated using transform techniques |
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Random signal |
a signal for which the signal value cannot be determined for a given value of time (i.e. noise from sound. It cannot be recreated and can't be expressed by a mathematical expression). Can be evaluated using probability and statistics. Modeling can be used so that random signals can be evaluated similarly to deterministic signals (using transform techniques). |
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Periodic signal |
A signal that can be written x(t) = x(t+T), where T is the period. |
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Angular frequency |
w = 2*pi*f = (2*pi)/T |
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Frequency |
f = 1/T |
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Period |
T = (2*pi)/w |
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Half angle formulas |
cos^2(A) = [1 + cos(2A)]/2 sin^2(A) = [1 - cos(2A)]/2 |
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Analog signals have ____ domain values and ____ co-domain values |
Continuous, continuous |
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Digital signals have ____ domain values and ____ co-domain values **************************************** |
Discrete, discrete (quantized) |
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Heaviside's Unit-Step signal: Expression Is it a power signal? Is it an energy signal? |
U(t) = 1, when t >= 0 and 0, when t =< 0 Neither a power (L^2) signal nor an energy signal. |
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Discrete Unit-Step expression |
U[n] = 1, when n >= 0 and 0, when n< 0 |
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Dirac Delta expression |
d(t) = 0, when t != 0 and infinite, when t = 0 |
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Impulse |
Equals 1 for a specified value of x and 0 for all other values |
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Impulse train |
Equals 1 for multiple specified values of x and 0 for all other values |
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Gating (gate function) |
Equals 1 over a period, T, and 0 for all other values; a rectangular shaped graph. Undoes convolution with an impulse train |
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Aliasing |
Overlapping in the time or frequency domain; y values add |
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Bandwidth |
Domain when frequency is on the x-axis |
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Euler's formula for complex numbers |
e^(st) = e^(sigma*t)[cos(wt) + jsin(wt)] Anything multiplied by j represents the imaginary part while anything not multiplied by j represents the real part. |
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Convolution with an impulse or impulse train results in ____. |
Shifting |
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Multiplication with an impulse or impulse train results in ____. |
Sifting (sampling) |
