Basic Operation on Signals Continuous-Time Signals

The signal is the actual physical phenomenon that carries information, and the function is a mathematical description of the signal.

Complex Exponentials & Sinusoids Signals can be expressed in sinusoid or complex exponential. g(t) = A cos (2Пt/T o +θ) = A cos (2Пf o t+ θ) = A cos (ω o t+ θ) g(t) = Ae (σ o +jω o )t = Ae σ o t [cos (ω o t) +j sin (ω o t)] Where A is the amplitude of a sinusoid or complex exponential, T o is the real fundamental period of sinusoid, f o is real fundamental cyclic frequency of sinusoid, ω o is the real fundamental radian frequency of sinusoid, t is time and σ o is a real damping rate. sinusoids complex exponentials

In signals and systems, sinusoids are expressed in either of two ways : a. cyclic frequency f form - A cos (2Пf o t+ θ) b. radian frequency ω form - A cos (ω o t+ θ) Sinusoids and exponentials are important in signal and system analysis because they arise naturally in the solutions of the differential equations.

Singularity functions and related functions In the consideration of singularity functions, we will extend, modify, and/or generalized some basic mathematical concepts and operation to allow us to efficiently analyze real signals and systems.

The Unit Step Function Precise GraphCommonly-Used Graph

The Signum Function Precise GraphCommonly-Used Graph The signum function, is closely related to the unit-step function.

The Unit Ramp Function The unit ramp function is the integral of the unit step function. It is called the unit ramp function because for positive t, its slope is one amplitude unit per time.

The Rectangular Pulse Function Rectangular pulse,

The Unit Step and Unit Impulse Function The unit step is the integral of the unit impulse and the unit impulse is the generalized derivative of the unit step

Graphical Representation of the Impulse The area under an impulse is called its strength or weight. It is represented graphically by a vertical arrow. Its strength is either written beside it or is represented by its length. An impulse with a strength of one is called a unit impulse.

Properties of the Impulse The Sampling Property The Scaling Property The sampling property “extracts” the value of a function at a point. This property illustrates that the impulse is different from ordinary mathematical functions. The Equivalence Property

The Unit Periodic Impulse The unit periodic impulse/impulse train is defined by The periodic impulse is a sum of infinitely many uniformly- spaced impulses.

The Unit Rectangle Function The signal “turned on” at time t = -1/2 and “turned back off” at time t = +1/2. Precise graph Commonly-used graph

The Unit Triangle Function The unit triangle is related to the unit rectangle through an operation called convolution. It is called a unit triangle because its height and area are both one (but its base width is not).

The Unit Sinc Function The unit sinc function is related to the unit rectangle function through the Fourier transform.

The Dirichlet Function The Dirichlet function is the sum of infinitely many uniformly-spaced sinc functions.

Combinations of Functions Sometime a single mathematical function may completely describe a signal (ex: a sinusoid). But often one function is not enough for an accurate description. Therefore, combination of function is needed to allow versatility in the mathematical representation of arbitrary signals. The combination can be sums, differences, products and/or quotients of functions.

Shifting and Scaling Functions Let a function be defined graphically by

1. Amplitude Scaling,

(cont…)

2. Time shifting, Shifting the function to the right or left by t 0

3. Time scaling, Expands the function horizontally by a factor of |a|

3. Time scaling, (cont…) If a < 0, the function is also time inverted. The time inversion means flipping the curve with the g axis as the rotation axis of the flip.

4. Multiple transformations A multiple transformation can be done in steps The order of the changes is important. For example, if we exchange the order of the time-scaling and time-shifting operations, we get: Amplitude scaling, time scaling and time shifting can be applied simultaneously.

Multiple transformations, A sequence of amplitude scaling, time scaling and time shifting

Differentiation and Integration Integration and differentiation are common signal processing operations in real systems. The derivative of a function at any time t is its slope at the time. The integral of a function at any time t is accumulated area under the function up to that time.

Differentiation

Integration

Even and Odd CT Functions Even FunctionsOdd Functions

Even and Odd Parts of Functions A function whose even part is zero is odd and a function whose odd part is zero is even.

Combination of even and odd function Function typeSumDifferenceProductQuotient Both evenEven Both oddOdd Even Even and oddNeither Odd

Two Even Functions Products of Even and Odd Functions

Cont… An Even Function and an Odd Function

Cont…

Two Odd Functions Cont…

Function type and the types of derivatives and integrals Function typeDerivativeIntegral EvenOddOdd + constant OddEven

Integrals of Even and Odd Functions

Signal Energy and Power The signal energy of a signal x(t) is All physical activity is mediated by a transfer of energy. No real physical system can respond to an excitation unless it has energy. Signal energy of a signal is defined as the area under the square of the magnitude of the signal. The units of signal energy depends on the unit of the signal. If the signal unit is volt (V), the energy of that signal is expressed in V 2.s.

Signal Energy and Power Some signals have infinite signal energy. In that case it is more convenient to deal with average signal power. The average signal power of a signal x(t) is For a periodic signal x(t) the average signal power is where T is any period of the signal.

Signal Energy and Power A signal with finite signal energy is called an energy signal. A signal with infinite signal energy and finite average signal power is called a power signal.

Basic Operation on Signals Discrete-Time Signals

Sampling a Continuous-Time Signal to Create a Discrete-Time Signal Sampling is the acquisition of the values of a continuous-time signal at discrete points in time x(t) is a continuous-time signal, x[n] is a discrete- time signal

Complex Exponentials and Sinusoids DT signals can be defined in a manner analogous to their continuous- time counter part g[n] = A cos (2Пn/N o +θ) = A cos (2ПF o n+ θ) = A cos (Ω o n+ θ) g[n] = Ae βn = Az n Where A is the real constant (amplitude), θ is a real phase shifting radians, N o is a real number and F o and Ω o are related to N o through 1/N 0 = F o = Ω o /2 П, where n is the previously defined discrete time. sinusoids complex exponentials

DT Sinusoids There are some important differences between CT and DT sinusoids. If we create a DT sinusoid by sampling CT sinusoid, the period of the DT sinusoid may not be readily apparent and in fact the DT sinusoid may not even be periodic.

DT Sinusoids 4 discrete-time sinusoids

DT Sinusoids An Aperiodic Sinusoid A discrete time sinusoids is not necessarily periodic

DT Sinusoids Two DT sinusoids whose analytical expressions look different, and may actually be the same. If then (because n is discrete time and therefore an integer), (Example on next slide)

Sinusoids The dash line are the CT function. The CT function are obviously different but the DT function are not.

The Impulse Function The discrete-time unit impulse (also known as the “Kronecker delta function”) is a function in the ordinary sense (in contrast with the continuous-time unit impulse). It has a sampling property, but no scaling property. That is,

The Unit Sequence Function

The Unit Ramp Function

The Rectangle Function

The Periodic Impulse Function

Scaling and Shifting Functions Let g[n] be graphically defined by

Scaling and Shifting Functions Amplitude scaling Amplitude scaling for discrete time function is exactly the same as it is for continuous time function

3. Time compression, K an integer > 1

4.

Differencing and accumulation The operation on discrete-time signal that is analogous to the derivative is difference. The discrete-time counterpart of integration is accumulation (or summation).

Even and Odd Functions

Combination of even and odd function Function typeSumDifferenceProductQuotient Both evenEven Both oddOdd Even Even and oddEven or OddEven or oddOdd

Products of Even and Odd Functions Two Even Functions

Cont… An Even Function and an Odd Function

Cont… Two Odd Functions

Accumulation of Even and Odd Functions

Signal Energy and Power The signal energy of a signal x[n] is

Signal Energy and Power Some signals have infinite signal energy. In that case It is usually more convenient to deal with average signal power. The average signal power of a signal x[n] is For a periodic signal x[n] the average signal power is

Signal Energy and Power A signal with finite signal energy is called an energy signal. A signal with infinite signal energy and finite average signal power is called a power signal.