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

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

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

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

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.

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.

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).

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<0 1 2 , 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).

Example with the Unit Step Function RC circuit with a switch

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.

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 > 1 2 1 2 , 𝑡 = 1 2 ∗ 1, 𝑡 < 1 2 Can be express using the unit step function. What is the area?

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.

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

Impulse Relationships Relating the impulse to the unit step function.

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

Impulses and other functions Sampling property of the impulse.

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

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.

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

Combinations of Functions Building Blocks Polynomial: g(t) = aNtN + aN-1tN-1 + ... + 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

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

Examples: Binary Amplitude Modulated CW

Examples: Binary Phase Modulated CW

Examples: Linear Chirp

Examples: 25% duty cycle PWM

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

Examples: Derivative & Integral of Previous

Examples: Derivative & Integral of Previous

Examples: Heartbeat

Examples: Earthquake

Examples: Bats

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

Even and Odd Even Signal Odd Signal Examples

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)

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

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

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.

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

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.