We use the term signal to mean a real or complex valued function of real variable(s) and denote the signal by x(t)
The variable t is called independent variable and the value x of t as dependent variable.
When t takes a vales in a countable set the signal is called a discrete time signal. For example
t ε {0, T, 2T, 3T, 4T,...}
t ε {....-1, 0 ,1,...}
t ε {1/2, 3/2, 5/2, 7/2,...}
For convenience of presentation we use the notation x[n] to denote discrete time signal. When both the dependent and independent variables take values in countable sets (two sets can be quite different) the signal is called Digital Signal.
When both the dependent and independent variable take value in continous set interval, the signal is called an Analog Signal.
Notation: When we write x(t) it has two meanings. One is value of x at time t and the other is the pairs (x(t), t) allowable value of t. By signal we mean the second interpretation. Notation for continous time signal {x(t)} denotes the continuous time signal. Here {x(t)} is short notation for {x(t), t ε I }
where I is the set in which t takes the value.
Notation for discrete time signal Similarly for discrete time signal we will use the notation {x(t)}, where {x(t)} is short for
{x(t), n ε I }.
Note that in {x(t)} and {x[n]} are dummy variables ie. {x[n]} and {x[t]} refer to the same signal. Some books use the notation x [.] to denote {x[n]} and x[n] to denote value of x at time
n. {x(t)} refers to the whole waveform,while x[n] refers to a particular value.
Most of the books do not make this distinction clean and use x[n] to denote signal and x[n0] to denote a particular value. . Discrete Time Signal Processing and Digital Signal Processing When we use digital computers to do processing we are doing digital signal processing. But most of the theory is for discrete time signal processing where dependent variable generally is continuous. This is because of the mathematical simplicity of discrete time signal processing. Digital Signal Processing tries to implement this as closely as possible. Thus what we study is mostly discrete time signal processing and what is really implemented is digital signal processing.
