Continuous Time Signal

A continuous Time Signal is one whose value is defined at every instant of time i.e it gives continuous values unlike discrete time signal



A continuous signal or a continuous-time signal is a varying quantity (a signal) whose domain, which is often time, is a continuum (e.g., a connected interval of the reals). That is, the function's domain is an uncountable set. The function itself need not be continuous. To contrast, a discrete time signal has a countable domain, like the natural numbers.
The signal is defined over a domain, which may or may not be finite, and there is a functional mapping from the domain to the value of the signal. The continuity of the time variable, in connection with the law of density of real numbers, means that the signal value can be found at any arbitrary point in time.
A typical example of an infinite duration signal is:
f(t) = sin(t)

Discrete Time Signal

A signal is said to be a Discrete time signal if it defines at discrete values of time i.e, it does not give value at every instant of time but at discrete time it gives value

discrete-time signal is a time series consisting of a sequence of quantities. In other words, it is a time series that is a function over a domain of discrete integers. Each value in the sequence is called a sample.
Unlike a continuous-time signal, a discrete-time signal is not a function of a continuous argument; however, it may have been obtained by sampling from a continuous-time signal. When a discrete-time signal is a sequence corresponding to uniformly spaced times, it has an associated sampling rate the sampling rate is not apparent in the data sequence, so may be associated as a separate data item.

Different Definitions of Signal

any nonverbal action or gesture that encodes a message; "signals from the boat suddenly stopped"

sign: communicate silently and non-verbally by signals or signs; "He signed his disapproval with a dismissive hand gesture"; "The diner signaled the waiters to bring the menu"

any incitement to action; "he awaited the signal to start"; "the victory was a signal for wild celebration"

bespeak: be a signal for or a symptom of; "These symptoms indicate a serious illness"; "Her behavior points to a severe neurosis"; "The economic indicators signal that the euro is undervalued"

an electric quantity (voltage or current or field strength) whose modulation represents coded information about the source from which it comes

notably out of the ordinary; "the year saw one signal triumph for the Labour party"

Introduction to the z-transform

The z-transform is useful for the manipulation of discrete data sequences and has acquired a new significance in the formulation and analysis of discrete-time systems. It is used extensively today in the areas of applied mathematics, digital signal processing, control theory, population science, economics. These discrete models are solved with difference equations in a manner that is analogous to solving continuous models with differential equations. The role played by the z-transform in the solution of difference equations corresponds to that played by the Laplace transforms in the solution of differential equations.

The function notation for sequences is used in the study and application of z-transforms. Consider a function defined for that is sampled at times , where is the sampling period (or rate). We can write the sample as a sequence using the notation . Without loss of generality we will set and consider real sequences such as, . The definition of the z-transform involves an infinite series of the reciprocals

Discrete Fourier series Transform

Suppose an audio signal sampled at 8kHz. This means that every succesive eighth of a milliseconds one makes a measurement of the intensity of the signal. For the remainder of this text, we will assume we're working on a sample of EIGHT measurements. Here is an exemple of such a sample : 50 mV 206 mV -100 mV -65 mV -50 mV -6 mV 100 mV -135 mV We could represent it by this graph :

Multiplication and convolution

Using the tool, review the transforms of the unit pulse function and the cosine function. For the moment it is best to view these using the magnitude and phase representation of the frequency domain.
Now switch to one of the 8ms segment of a cosine or sine waveforms. You should observe that the frequency domain plot is some form of combination of the two types of signal. Strictly speaking, the time domain signal is the multiplication of a unit pulse of 8ms duration delayed by 4ms, and a cosinusoid or sinusoid waveform of the selected frequency. The frequency domain transform is then the addition of two sa functions which have been shifted in frequency. Notice where the highest peaks are and you should observe that these correspond with the frequency of the sine or cosine signal. What has happened is that in the frequency domain the sa function from the unit pulse and the two impulses from the sine or cosine function have been convolved together. This is an example of the general rule that multiplication in the time domain equates to convolution in the frequency domain.
You can reconstruct the two constituent waveforms by shifting the frequency response of the 8ms unit pulse to 500Hz, and to -500Hz.You should find that the real component of the two shifted signals are the same, but that the quadrature components are the complement of each other. Thus when they are summed together, the result is a signal with a real component and a zero quadrature component.
In fact an equivalent rule also holds that convolution in the time domain equates to multiplication in the frequency domain. Thus, for example, a complex phasor in the frequency domain multiplied by a given signal's transform produces a time domain function where an impulse is convolved with signal. This is precisely what is happening when the delay value is being altered.

Fourier Transforms

The Fourier transform defines a relationship between a signal in the time domain and its representation in the frequency domain. Being a transform, no information is created or lost in the process, so the original signal can be recovered from knowing the Fourier transform, and vice versa.
The Fourier transform of a signal is a continuous complex valued signal capable of representing real valued or complex valued continuous time signals.

The tool allows you to view these complex valued signals as either their real and quadrature (also known as imaginary) components separately, or by a magnitude and phase representation. You may switch between these two representations at any point. Mathematically switching between the two representations for a given complex value can be expressed as