Properties

Continuous (CT) vs discrete (DT)

Periodicity

CT \(x(t + nT) = x(t)\) for all \(t\)
- Frequency \(f = \frac{1}{T}\) where \(T\) is the period
DT \(x[n + kN] = x[n]\) for all \(n\)

The domain of a periodic signal is \((-\infty, \infty)\)

See example exercises here

Deterministic vs random

If the signal can be represented by an equation (ex: sin, cos vs random data)

Symmetry - Even, Odd

Even: \(x_e (t) = x_e (-t)\)

Odd: \(x_o (t) = -x_o (-t)\)

Say we have two functions \(f, g\). Then

Function types	\(f+g\)	\(f \times g\)
\(f, g\) even	Even	Odd
\(f, g\) odd	Odd	Even
one even, one odd	Neither	Odd

Any function can be broken down into an even and odd component:
\[x(t) = x_e (t) + x_o (t)\] \[x_e(t) = \frac{1}{2} (x(t) + x(-t))\] \[x_o(t) = \frac{1}{2} (x(t) - x(-t))\]

Any complex signal (with imaginary number) can be broken down into a conjugate-symmetric and conjugate-antisymmetric (CS, CA):
\[x(t) = x_{CS} (t) + x_{CA} (t)\] \[x_{CS} (t) = \frac{x(t) + x^* (-t)}{2}\] \[x_{CA} (t) = \frac{x(t) - x^* (-t)}{2}\]
This is useful for functions with complex values, like \(e^{jt}\)

Energy

\[E = \int_{T_1}^{T_2} \left| {x(t)} \right|^2 dt\]

\[E = \sum_{n=N_1}^{N_2} \left| {x[n]} \right| ^2\]

For total energy, the time interval is \((-\infty, \infty)\)

Power

\[P_x = \frac{1}{T_0} \int_{t_1}^{t_1 + T_0} \left| {x(t)} \right|^2 = \frac{1}{T_0} E\] \[P_x = \frac{1}{N_0} \sum_{n_1}^{n_1 + N_0 -1} \left| {x[n]} \right| ^2 = \frac{1}{N_0} E\]

Which is basically the average of energy in some time (the above is average power).

A signal with finite energy has 0 power, and a signal with finite power has \(\infty\) energy

Operations

Time shifting \(x(t-T)\)

Time scaling \(x(kt)\)

For DT, time scaling \(x[kn]\) creates or deletes values from the signal:

if (k > 1) => y[n] = x[kn] retains values only from the scale (x[0], x[M], x[2M]…), and other values in between are removed.

if (k < 1) => y[n] = x[kn] interpolates from original values to create the new ones

Time reversal \(x(-t)\) mirror y-axis

Combined operations \(x(at + b) = x(a(t + \frac{b}{a}))\), first scale then shift

Elementary Signals

Unit step \(u(t) = 1\) if \(t > 0\), otherwise 0

Rectangular pulse \(rect(\frac{t}{\tau}) = 1\) if \(|t| < \frac{\tau}{2}\)

Signum (sign) \(sgn(t)\) is 1 times the sign of \(t\), or 0 if \(t = 0\)

Ramp \(r(t) = tu(t)\)

Sines \(A\sin(\omega t + \theta)\)

Angular frequency \(\omega = 2\pi f\)

Sinc \(sinc(kt) = \frac{\sin(kt)}{kt}\)

Exponential \(e^{j\omega t}\)

Unit impulse \(\delta (t) = 1\) only at \(t=0\), and 0 otherwise

Unit impulse \(\delta (t)\)

\(\delta (t) = \delta (-t)\) (even)
Given a function \(f(t)\):
- The product \(f(t) \delta(t) = f(0) \delta(t)\)
- General form: \(f(t) \delta(t - t_0) = f(t_0) \delta(t - t_0)\)
The integral \(\int_{-\infty}^{\infty}\delta (t) = 1\)

\(u(t)\) and \(\delta (t)\) are related:

\[\delta (t) = \frac{du(t)}{dt}\] \[u(t) = \int_{-\infty}^{t}\delta(\tau)d\tau\]

Problems

Type 1: Continuous Periodic Signals

Given:

Some continuous time equation \(x(t)\)

Find:

The fundamental period \(T\), if periodic

Process:

Solve for \(nT\):

\[x(t + nT) = x(t)\]

The \(t\) should cancel out in the process, leaving an equation in terms of \(nT\).

IF (\(t\) remains, or \(T \leq 0\)):
=> \(x(t)\) is non-periodic

IF x(t) is of type sin, cos:

Rewrite in structure \(\sin(\omega t + 2\pi k) = \sin(\omega t + \omega nT)\)
Get \(2\pi k = \omega nT\)
Let \(k = n\), then \(T = \frac{2\pi}{\omega}\)

IF (\(x(t) = a(t) + b(t)\) where \(a,\ b\) are both periodic):
=> \(T_x = LCM(T_a,\ T_b)\) (least common multiple)
NOTE: \(T_x\) from the LCM may be irrational (\(\pi, \sqrt{2}\)), but the multipliers of \(T_a,\ T_b\) MUST be integers to reach the LCM

If T applies for all n (where n is an int):
=> \(x(t)\) is periodic, with period T

Type 1 Examples

Answers in the bottom.

\[e^{it}\] \[e^t\] \[ln \mid t \mid\] \[\sin \sqrt{2}t\] \[\sin t + \sin \pi t\] \[\sin{t^2}\] \[e^{j(2t+7)}\] \[5\cos(3\pi t) + 3\cos(5\pi t)\]

Type 2: Discrete Periodic Signals