CT/DT Signals and Properties

Study Notes

Important Continuous-Time (CT) and Discrete-Time (DT) functions include the unit impulse, unit step, exponential (complex or real), sine/cosine, and Euler's formula.
Common properties for CT and DT signals are: even/odd/neither, periodic/aperiodic, power/energy/neither.
For periodic signals, the fundamental period is the smallest positive real number T (CT) or positive integer N (DT) for which the signal repeats.
For periodic signals, the fundamental frequency is the reciprocal of the fundamental period.
The sum and product of two periodic signals are also periodic.
To calculate the energy/power of CT/DT signals, specific formulas are used depending on whether the signals are continuous or discrete in time.

Properties of CT/DT systems include Stability, Invertibility, Memory, Causality, Time Invariance, and Linearity.
You can determine if a system possesses these properties by analyzing the system's impulse response function and applying definitions.

Convolution is defined as the integral (CT) or summation (DT) of the product of two functions shifted in time.
Convolution is used to find the output y(t) of a system when the input x(t) and impulse response h(t) are known.
Key properties of convolution include commutativity, distributivity, associativity, and time-shift.
There are convolution tables that can be used to simplify the calculation of specific convolutions.

The impulse response function describes the output of a system when the input is a unit impulse.
For Linear Constant Coefficient Differential Equations (LCCDEs) in CT, the impulse response is found by solving a homogeneous differential equation with an initial condition determined by the input signal and the system's coefficients.
The homogeneous solution can be found by solving for the roots of the characteristic equation associated with the LCCDE. There can be real roots, complex roots, or repeated roots.
The constants in the homogeneous solution are determined by using auxiliary conditions.

Advance form and delay form are two representations of LCCDEs.
The LCCDE can be expressed using the shift operator E to simplify notation and manipulation.
To solve for the impulse response function h(n) of a DT system, we need to find the homogeneous solution y_h(n).
The homogeneous solution can be found by solving for the roots of the characteristic equation associated with the LCCDE using the shift operator representation.
The constants in the homogeneous solutions are determined by iteratively calculating the values of the impulse response h(n) starting with the initial conditions.

The impulse response function h(n) is the output of a DT system when the input is a unit impulse.
For LCCDEs in DT, the impulse response function is determined by solving a homogeneous difference equation with initial conditions guided by the input and coefficients.
The homogeneous solution is found by finding the roots of the characteristic equation and using them to construct the general solution form.
The constants in the general solution can be determined iteratively.

Convolution can be visualized using a graphic approach by shifting and multiplying the input signal and the impulse response function.
The convolution process involves sliding the impulse response function across the input signal, multiplying the overlapping parts, and summing the results.
This graphical method provides a visual understanding of how convolution operates.