Chapter 3

Question 1

What is Time shifting (also called translation)

Answer 1

maps the input function x to the output function y as given by y(t) = x(t −b).

If b > 0, y is shifted to the right by |b|, relative to x (i.e., delayed in time).
If b < 0, y is shifted to the left by |b|, relative to x (i.e., advanced in time).

Question 2

What is Time reversal (also known as reflection)

Answer 2

maps the input function x to the output function y as given by y(t) = x(−t).

Question 3

What is Time compression/expansion (also called dilation)

Answer 3

maps the input
function x to the output function y as given by
y(t) = x(at), where a is a strictly positive real number.

If a > 1, y is compressed along the horizontal axis by a factor of a, relative
to x.
If a < 1, y is expanded (i.e., stretched) along the horizontal axis by a factor
of 1
a
, relative to x.

Question 4

In what order would you do the time scaling and time shifting of
y(t) = x(at −b)

Answer 4

You would first time shift by b, then scale by a.

Or if you factor the a out you get y(t) = x(a(t −(b/a))) and would first scale by a, then shift by b/a

Question 4

What is Time scaling

Answer 4

The combination of Time compression and Time reversal, maps the input function x to the output function y as given by y(t) = x(at), where a is a nonzero real number.

Question 5

What is amplitude scaling?

Answer 5

maps the input function x to the output function y as
given by y(t) = ax(t),
where a is a real number.

Question 6

What is amplitude shifting?

Answer 6

maps the input function x to the output function y as
given by y(t) = x(t) +b,
where b is a real number

Question 7

In what order would you do the amplitude scaling and amplitude shifting of
y(t) = ax(t)+b

Answer 7

first amplitude scaling x by a, and then amplitude shifting the resulting
function by b

Or if you factor out the a you get y(t) = a(x(t) + (b/a)) where it would then be first amplitude shifting x by b/a, and then amplitude scaling the resulting
function by a

Question 8

What is the resulting symmetry from an addition of all different combinations of even and odd symmetry

Answer 8

1) The sum of two even functions is even.
2) The sum of two odd functions is odd.
3) The sum of an even function and odd function is neither even nor odd,
provided that neither of the functions is identically zero

Question 9

What is the resulting symmetry from an multiplication of all different combinations of even and odd symmetry

Answer 9

1) The product of two even functions is even.
2) The product of two odd functions is even.
3) The product of an even function and an odd function is odd.

Question 10

How can a function be broken down into its even and odd parts? And what are the formulas of those even and odd parts?

Answer 10

X(t) = Xe(t) +Xo(t)
where:
Xe(t) = (1/2)[x(t)+x(−t)] and Xo(t) = (1/2)[x(t)−x(−t)].

Xe(t) is also labeled Even(x)
Xo(t) is also labeled Odd(x)

Question 11

For two periodic functions x1 and x2 with fundamental periods T1 and T2, respectively, and the sum y = x1 +x2: How do you find if y is periodic? And if it is, how do you find the fundamental period?

Answer 11

The sum y is periodic if and only if the ratio T1/T2 is a rational number (i.e., the quotient of two integers).

If y is periodic, its fundamental period is rT1 (or equivalently, qT2, since
rT1 = qT2), where T1/T2 = q/r and q and r are integers and coprime (i.e.,
have no common factors). (Note that rT1 is simply the least common
multiple of T1 and T2.)

Question 12

What is a right sided function?

Answer 12

For some (finite) real constant t0, the following condition holds: x(t) = 0 for all t < t0 (i.e., x is only potentially nonzero at t0 or to the right of t0).

Question 13

What is a causal function?

Answer 13

x(t) = 0 for all t < 0 (i.e., x is only potentially nonzero at 0 or to the right of 0).

Question 14

What is a left sided function?

Answer 14

For some (finite) real constant t0, the following condition holds: x(t) = 0 for all t > t0 (i.e., x is only potentially nonzero at t0 or to the left of t0).

Question 15

What is an anticausal function?

Answer 15

x(t) = 0 for all t > 0 (i.e., x is only potentially nonzero at 0 or to the left of 0).

Question 16

What is a finite duration (or time limited) function?

Answer 16

A function that is both left sided and right sided

Question 17

What is a bounded function?

Answer 17

a function is bounded if there exists some (finite) positive real constant A such that |x(t)| ≤ A for all t
(i.e., x(t) is finite for all t).

Question 18

What is the energy of a function?

Answer 18

E = integral_from(−∞,∞) of(|x(t|^(2)dt).

Question 19

What is an energy signal

Answer 19

A signal with finite energy

Question 20

What is the average power?

Answer 20

Average power P contained in the function x is given by
P = lim_from(T→∞)of((1/T)integral_from(−T/2, T/2)of(|x(t)|^(2)dt)

Question 21

What is a power signal?

Answer 21

A signal with (nonzero) finite average power

Question 22

What is a real sinusoidal function? What is its periodicity and frequency? For Georgia

Answer 22

A function of the form
x(t) = Acos(ωt +θ),
where A, ω, and θ are real constants.
Such a function is periodic with fundamental period
T = (2π)/|ω| and fundamental frequency |ω|.

Question 23

What is a complex exponential function?

Answer 23

a function of the form
x(t) = Ae^(λt), where A and λ are complex constants.

Question 23

Info on general complex functions

Answer 23

In the most general case of a complex exponential function x(t) = Ae^(λt), A and λ are both complex. Letting A =|A|e^(jθ) and λ = σ+ jω (where θ, σ, and ω are real), and using Euler’s relation, we can rewrite x(t) as x(t) = |A|e^(σt)cos(ωt +θ) + j|A|e^(σt)sin(ωt +θ). Thus, Re{x} and Im{x} are each the product of a real exponential and real sinusoid. One of three distinct modes of behavior is exhibited by x(t), depending on the value of σ. 1) If σ = 0, Re{x} and Im{x} are real sinusoids. 2) If σ > 0, Re{x} and Im{x} are each the product of a real sinusoid and a growing real exponential. 3) If σ < 0, Re{x} and Im{x} are each the product of a real sinusoid and a decaying real exponential.

Question 23

What is a complex sinusoidal function?

Answer 23

A complex sinusoidal function is a special case of a complex exponential x(t) = Ae^(λt), where A is complex and λ is purely imaginary (i.e., Re{λ} = 0). That is, a complex sinusoidal function is a function of the form x(t) = Ae^(jωt), where A is complex and ω is real. By expressing A in polar form as A = |A|e^(jθ) (where θ is real) and using Euler’s relation, we can rewrite x(t) as x(t) =|A|cos(ωt +θ) + j|A|sin(ωt +θ) Thus, Re{x} and Im{x} are the same except for a time shift. Also, x is periodic with fundamental period T = (2π)/|ω|and fundamental frequency |ω|.

Question 23

What is a real exponential function?

Answer 23

A special case of a complex exponential x(t) = Ae^(λt), where A and λ are restricted to be real numbers. A real exponential can exhibit one of three distinct modes of behavior, depending on the value of λ. 1) If λ > 0, x(t) increases exponentially as t increases (i.e., a growing exponential). 2) If λ < 0, x(t) decreases exponentially as t increases (i.e., a decaying exponential). 3) If λ = 0, x(t) simply equals the constant A.

Question 23

What is the signum function?

Answer 23

sgn(t) = {(1 if t > 0), (0 if t = 0), (−1 if t < 0)}

Question 24

What is a unit step function? (also known as the Heaviside function)

Answer 24

u(t) = {(1 if t ≥ zero), (0 otherwise)}

Question 24

What is a rectangular function? (also called the unit-rectangular pulse function)

Answer 24

rect(t) = {(1 if -(1/2) ≤ t < (1/2)), (0 otherwise)}

Question 25

What is the cardinal sine function?

Answer 25

sinc(t) = sin(t)/t. Note: By l’Hopital’s rule, sinc 0 = 1.

Question 26

How do you represent the delta function as a limit?

Answer 26

δ(t)=lim_from(ε→0)of(gε(t)). where gε(t) = {(1/ε if|t|< ε/2), (0 otherwise)}

Question 26

What is the unit-impulse function? (also known as the Dirac delta function or delta function)

Answer 26

δ(t) = 0 for t not equal to 0 and integral_from(−∞,∞)of( δ(t)dt) = 1.

Question 27

What is the equivalence property?

Answer 27

For any continuous function x and any real constant t0, x(t)δ(t −t0) = x(t0)δ(t −t0)

Question 28

What is the sifting property

Answer 28

For any continuous function x and any real constant t0, integral_from(−∞,∞)of( x(t)δ(t −t0)dt) = x(t0).

Question 29

How do you collapse a piecewise function into a single expression?

Answer 29

By using unit step functions. for every statement of the piecewise function (i.e. x(t) = k if a ≤ t ≤ b) can be represented by k[u(t - a) - u(t - b)]. Adding all the statements together makes the single expression that is equivalent to the original piecewise function