Discrete-Time Signals: Definition, Representation, and Sampling
Learn about discrete-time signals, sequences of numbers defined at distinct points in time. This guide explains their representation, how they are obtained from continuous signals through sampling, and their importance in digital signal processing.
Discrete-Time Signals in Discrete Mathematics
What are Discrete-Time Signals?
A discrete-time signal is a sequence of numbers defined only at specific, distinct points in time. Unlike continuous signals (like a smoothly flowing sound wave), discrete signals are like a series of snapshots taken at regular intervals.
Representing Discrete-Time Signals
We represent discrete-time signals using the notation x[n], where 'n' is an integer representing the time index (often starting at 0, but could start at any other integer, and range from negative infinity to positive infinity), and x[n] is the value of the signal at that time point. A discrete-time signal can be represented as:
x = {x[n]}, -∞ < n < ∞
Obtaining Discrete-Time Signals from Continuous Signals
We often get discrete-time signals by sampling continuous-time signals. This involves taking measurements of the continuous signal at regular intervals. The sampling formula is:
x[n] = xa(nT), -∞ < n < ∞
Where xa(t) is the continuous-time signal, T is the sampling period (the time between samples), and x[n] is the value of the discrete signal at the nth sample.
The sampling frequency (fs) is the number of samples per unit time and is related to the sampling period by fs = 1/T.
Graphical Representation
Discrete-time signals are typically plotted using a stem plot. Each point on the plot represents a sample (x[n]) at a specific time index (n). The values of x[n] are undefined for non-integer values of n.
(An example of a stem plot of a discrete-time signal would be included here.)
Representing Discrete-Time Signals
Discrete-time signals can be shown in several ways:
- Functional Representation: Defining x[n] explicitly for each integer value of n.
- Tabular Representation: A table listing each value of n and its corresponding x[n].
- Sequence Representation: A list of the signal values, with an indicator showing the sample corresponding to n = 0 (often marked with an arrow).
(Illustrative examples using each of these representations are provided in the original text and should be added here.)
Operations on Discrete-Time Sequences
We can perform various operations on discrete-time signals:
- Sum: {cn} = {an} + {bn} => cn = an + bn
- Product: {cn} = {an} * {bn} => cn = an * bn
- Scalar Multiplication: {cn} = k{an} => cn = k * an
Special Discrete-Time Signals
Unit Sample Sequence (δ[n])
The unit sample sequence (also called the impulse sequence) is a fundamental signal. It's defined as:
δ[n] = 1 for n = 0; δ[n] = 0 otherwise.
Arbitrary discrete-time signals can be represented as a sum of scaled and shifted unit sample sequences.
(An example showing how an arbitrary sequence can be expressed using the unit sample sequence would be included here.)
Exponential Sequences
Exponential sequences have the form x[n] = Aαn, where A and α are constants. The behavior of the sequence depends on α:
- 0 < α < 1: Decreasing sequence.
- -1 < α < 0: Alternating, decreasing sequence.
- |α| > 1: Growing sequence.
Complex Exponential Sequences
A complex exponential sequence x[n] = |A|ej(ω0n + φ) is periodic when |α| = 1. It can also be expressed using sine and cosine functions.
Conclusion
Discrete-time signals are fundamental in digital signal processing and various other applications. Understanding their representation, operations, and special types is essential for working with digital data.