Continuous-Time and Discrete-Time Signals
If a signal value is defined for every instant of time in a considered interval however small the interval is, it is called a “continuous-time signal”. A discrete-time signal is derived from a continuous-time signal through a process called uniform sampling, hence it is defined for discrete values of the time.
For a discrete-time signal module f(n), the argument ‘n’ must be an integer, in contrast, the argument ‘t’ of the continuous-time signal can be any real value.
If a continuous-time signal time signal f(t) is uniformly sampled to form a discrete-time signal f(nTs) such that t=nTs
f(nTs) = f(t), n = 0, ±1, ±2, ±3,….
where Ts is the sampling period (sec/sample).
If Ts = 1 sec/sample, then f(nTs)=f(n)
The below figure shows a continuous-time signal f(t)=sin t and its discrete-time version f(nTs)
The continuous-time signal f(t) can be defined either in the form of a mathematical equation or in a graphical form. On the other hand, along with mathematical form, and a graphical representation, the discrete-time signal f(n) can be represented in the set of values as shown below.
Here the arrow indicates the amplitude of the sample at the origin, and the values right to the arrow represent the amplitude of the sample for the positive values of ‘n’ and the values to its left will be the sample amplitudes for the negative values of ‘n’.
Even and Odd Signals
If f(t) is a continuous-time signal, that is symmetric about the vertical axis such that it appears visually identical to its time-reversal version, such signal is known as an even signal.
f(−t) = f(t)
For a discrete-time signal
f(−n) = f(n)
If f(t) is a continuous-time signal, it will be called an odd signal, if it is anti-symmetric about its vertical axis, such that
For a discrete-time signal to be an odd signal
But in practice, most of the signals are neither even nor odd, for such signals we can find even and odd parts such signals to exploit the symmetry in the analysis of signals and systems. If
f(t) is any continuous-time signal that is neither an even signal nor an odd signal, then
- Even part of the signal is
- The odd part of the signal is
Then f(t) is representable as the sum of its even component (fe(t))and its odd component(fo(t)).
f(t) = (fe(t))+(fo(t))
Periodic and Aperiodic Signals
If any signal has a definite pattern, that repeats itself at a regular interval, such signal is known as a periodic signal. If it does not have such a pattern, then it is known as an aperiodic signal. If f(t) is a continuous-time periodic signal, then
Where ‘T’ is the fundamental time period of the signal f(t), and ‘k’ is an integer.
Where, ωis the angular frequency, ω=2πf, and don’t be confused with signal f(t) and frequency ‘f’.
If f(n) is a discrete-time signal,
f(n) = F(n+N)
Where ‘n’ is an integer and ‘N’ is the fundamental time period. Here N is the number of samples taken by the signal to repeat the same pattern, it is an integer, in contrast ‘T’ can be any real value for a continuous-time signal.
Consider the following example for a better understanding.
Energy Signals and Power Signals
If the total energy of the signal is finite, such signals are called energy signals, power of such signals is zero. The signals having infinite energy and finite power are known as power signals. To have precise clarity on this we have to know how to calculate the power and energy of the signal, for this understanding consider an electric circuit having a resistor of resistance (R), applied voltage v(t), and the current flowing through the resistor i(t). The instantaneous power is given by
To calculate the normalized power let us assume the resistance R=1
By accumulating the instantaneous power over an interval [Equation] to [Equation], we can obtain the energy dissipated in that interval E(t).
Hence by using this example we can generalize the energy of any continuous-time signal f(t)
Total normalized energy
Then we will calculate the power
Hence we have to calculate the energy first, if the energy of the signal is finite then it is an energy signal, the power of an energy signal is almost negligible hence we can consider it as zero.
If the energy of the signal is infinite, then we can’t calculate the power of such a signal directly since
For such signals, we have to calculate the power indirectly as shown below
If the signal has infinite energy and finite power such signal can be called the power signal. If the signal has infinite energy and infinite power, such signals are neither energy nor power signals.
- In general, the power signals are having an infinite duration with their value tending to a non-zero constant value as time tends to infinity. Hence periodic signals are power signals in general.
- The signals having infinite duration with their values also tend to infinity, such signals are called neither energy nor power signals.
If f(n) is a discrete-time signal, then the energy and power of the signal are given by
Deterministic and Random Signals
If a signal’s future value is known exactly at present, such signal is called a deterministic signal.
Example: f(t) = 3t + 6
At t=0; f(t) = 3(0) + 6 = 6 (present value)
At t=10; f(t) = 3(10) + 6 = 36 (future value)
If the future value of the signal cannot be determined at present, such signals are called random signals, such signals can only be expected or estimated.
Causal and Non-causal Signals
If f(t) is a signal such that
f(t)= 0; t<0
Then f(t) is known as a causal signal.
If f(t) is a signal that has a defined value in the interval t<0; such signals are known as non-causal signals.
Analog and Digital Signals
Concerning time, the signals are classified into continuous-time signals and discrete-time signals. Similarly based on the value the signals are classified into analog and digital. If the value of the signal can be any real value in its dynamic range such signals are known as analog signals. Whereas if the signal is allowed to consider the specific values in its dynamic range such signals are known as digital signals.
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