1. Sinusoidal Signal Described using three parameters Equation
Amplitude
Frequency or period of oscillation
Phase or time offset
Equation
𝑔 𝑡 =𝐴∙ cos 2𝜋 𝑓 0 𝑡+𝜃
Variations

2. Real Exponential Signal
Described using two parameters
Amplitude
Rate of decay or growth
Equation
𝑔 𝑡 =𝐴∙ 𝑒 𝜎 0 𝑡
Variations

3. Complex Exponential Signal
Described using three parameters
Amplitude
Rate of decay or growth
Frequency of oscillation
Equation
𝑔 𝑡 =𝐴∙ 𝑒 𝜎 0 +𝑗 𝜔 0 𝑡
Variations

4. Complex Exponential Signal
The complex exponential can be split into real and imaginary components that are expressed as combination of the real exponential and sinusoidal signals.
Equation
𝑔 𝑡 =𝐴∙ 𝑒 𝜎 0 +𝑗 𝜔 0 𝑡 =𝐴∙ 𝑒 𝜎 0 𝑡 ∙ cos 𝜔 0 𝑡 +𝑗 sin 𝜔 0 𝑡
Angular Rotation in the complex plane as time increases

5. Complex Exponential Signal
Recovering sin and cos from the complex exponential
Equation
𝑔 1 𝑡 =𝐴∙ 𝑒 𝑗 𝜔 0 𝑡 =𝐴∙ cos 𝜔 0 𝑡 +𝑗 sin 𝜔 0 𝑡
𝑔 2 𝑡 =𝐴∙ 𝑒 −𝑗 𝜔 0 𝑡 =𝐴∙ cos 𝜔 0 𝑡 −𝑗 sin 𝜔 0 𝑡
Evaluate g1 + g2 and g1 – g2
𝑔 1 + 𝑔 2 =2𝐴∙ cos 𝜔 0 𝑡
𝑔 1 − 𝑔 2 =𝑗2𝐴∙ sin 𝜔 0 𝑡
Which also gives relationships for sinω0t and cosω0t.

6. Discontinuities
There are a class of related functions that represent a discontinuities.
lim 𝜀→0 𝑔 𝑡 0 −𝜀 ≠ lim 𝜀→0 𝑔 𝑡 0 +𝜀
The mathematical description of these functions require some type of limit or conditional inequality.
These are useful because they can be used to model switching and sampling events as well.

7. Signum
The signum function can be interpreted as a test that returns the sign of the input.
𝑠𝑔𝑛 𝑡 = −1, 𝑡<0 0, 𝑡=0 1, 𝑡>0
All other discontinuous function can be derived from this function (or visa versa).

8. Unit Step Function
The unit step function signifies activation at time zero.
It is zero for time less than zero and unity for time greater than zero.
𝑢 𝑡 = 0, 𝑡< , 𝑡= 0 ∗ 1, 𝑡>0
In some cases, t=0 is ignored or included in one of the other cases.
Express in terms of sgn(t).

9. Example with the Unit Step Function
RC circuit with a switch

10. Unit Ramp Function
As the name implies, the unit ramp starts increasing from zero at time zero with a slope of one.
ramp 𝑡 = 0, 𝑡≤0 𝑡, 𝑡>0
Can be express using the unit step function. Again.

11. Unit Rectangle Function
The unit rectangle function is similar to the unit step function except it is on for the time interval -1/2 to +1/2.
rect 𝑡 = 0, 𝑡 > , 𝑡 = 1 2 ∗ 1, 𝑡 < 1 2
Can be express using the unit step function.
What is the area?

12. Impulse Function
The impulse function has two important characteristics:
Has a value of zero at every point except t=0.
Has an area of unity.
𝛿 𝑡 =0, 𝑡≠0 and 𝑡 1 𝑡 2 𝛿 𝑡 𝑑𝑡 = 1, 𝑡 1 <0< 𝑡 1 0, otherwise
As a result, the impulse function, δ(t), is somewhat difficult to express directly.
Point 2 above, seems to indicate some relationship between the impulse and the unit rectangle function.

13. Impulse Function
Deriving the impulse function using limits and the unit rectangle function.
Graphical Representation.

14. Impulse Relationships
Relating the impulse to the unit step function.

15. Impulses and other functions
Multiply a function by an impulse.
Multiply a function by a shifted impulse

16. Impulses and other functions
Sampling property of the impulse.

17. Impulses and other functions
Scaling property of the impulse.
Related to the unit rectangle function (first).

18. The Impulse Train
A series of impulse functions shifted by multiples of T.
𝛿 𝑇 𝑡 = 𝑛=−∞ ∞ 𝛿 𝑡−𝑛𝑇
Apply scaling/time shift to the impulse train.

20. Manipulating Functions
We can manipulate function to represent or model more interesting signals.
Think of the previous definitions as building blocks.
Combinations of multiple functions are constructed using arithmetical operations (+,-,×,÷).
Be careful when dividing by zero.
Functions can be scaled and shifted to make them more useful.
Be careful when integrating.
Integration and differentiation can be used to further manipulate functions.
Nested functions

21. Combinations of Functions
Building Blocks
Polynomial: g(t) = aNtN + aN-1tN a2t2 + a1t + a0
Sinusoidal Signal
Exponential (real and complex)
Signum
Unit Step, Unit Ramp, and Unit Rectangle
Impulse and impulse Train
Operations
Arithmetic
Time Shifting
Derivatives and Integrals
Functions of functions
Creativity

22. Examples: Binary (...,1,1,0,1,0,0,0,1,0,1,0,...)

23. Examples: Binary Amplitude Modulated CW

24. Examples: Binary Phase Modulated CW

25. Examples: Linear Chirp

26. Examples: 25% duty cycle PWM

27. Examples: Δ, 50% DC, 1 s period, +/-5 Amp

28. Examples: Derivative & Integral of Previous

29. Examples: Derivative & Integral of Previous

30. Examples: Heartbeat

31. Examples: Earthquake

32. Examples: Bats

33. Properties of Signals Symmetry Periodic Energy and Power
Even and odd signals
Even and odd Components
Sums and products of even and odd signals
Periodic
Fundamental period
Combinations of periodic signals
Energy and Power

34. Even and Odd
Even Signal
Odd Signal
Examples

35. Even and Odd Components
A signal, g(t), can be broken down into an even, ge(t), and odd, go(t) components – where
g(t) = ge(t) + go(t)
with
ge(t) = ge(-t)
go(t) = -go(-t)

36. Even and Odd Components
Even component: ge(t)
Odd component: go(t)

37. Combining Even and Odd Signals
Addition (subtraction): Stays the same
Multiplication
ge(t) * fe(t) =
ge(t) * fo(t) =
go(t) * fe(t) =
go(t) * fo(t) =
Derivate, Integral

38. Periodicity A signal is periodic if:
g(t) = g(t-nT) when T is a constant and n is an integer
The mimimul value of T is considered the fundamental period, T0 (f0 = 1/T0).
Sums of periodic signals are also periodic.

39. Power Power 𝑃 𝑥 𝑡 = 𝑥 𝑡 2 A periodic signal has an average power
𝑃 𝑥 𝑡 = 𝑥 𝑡 2
A periodic signal has an average power
𝑃 𝑥 𝑎𝑣𝑔 = 1 𝑇 𝑜 𝑡 𝑜 𝑡 𝑜 + 𝑇 𝑜 𝑥 𝑡 2 𝑑𝑡
The average power of an aperiodic signal is taken as the limit T → ∞.
𝑃 𝑥 𝑎𝑣𝑔 = lim 𝑇→∞ 1 𝑇 𝑡 𝑜 𝑡 𝑜 +𝑇 𝑥 𝑡 2 𝑑𝑡

40. Energy The energy is the integral of power. 𝐸 𝑥 = −∞ ∞ 𝑥 𝑡 2 𝑑𝑡
𝐸 𝑥 = −∞ ∞ 𝑥 𝑡 2 𝑑𝑡
A signal with finite Energy is called an energy signal.
What is the average power of an energy signal?
A signal with infinite Energy and finite average power is called a power signal.