Sampling Theorem
When the sampling frequency fs is larger than or equal to twice the highest frequency component of the message signal, a continuous time signal can be represented in its samples and retrieved.
fs ≥ 2fm
fm-> is band limit frequency
fs -> sampling frequency
Let consider a signal x(t) and the impulse train δ(t – nTs)
Input signal
impulse train -> δ(t – nTs)
Impulse Train
now Y(t) = x(t) . δ(t – nTs) …… eq 1
Sampled signal
taking Fourier transform of the first equation.
Ys(f) = X(F) ✻ Fs Σ δ(f – nfs)
Ys(f) = fs Σ X(f – nfs)
on plotting the following Ys(f) with frequency.
Output signal
Here, to avoid the overlapping and to get perfect sample :
fs ≥ 2fm .
Aliasing and Anti-Aliasing
When the fs < 2fm then overlapping of the sampling takes place called Aliasing effect. An anti-aliasing filter eliminates any potential under-sampled frequencies from the signal by examining the user-specified sampling frequency.
Aliasing effect
LTI System
Systems that are both linear and time-invariant are known as linear time-invariant systems, or LTI systems for short. When a system’s outputs for a linear combination of inputs match the outputs of a linear combination of each input response separately, the system is said to be linear. Time-invariant systems are ones whose output is independent of the timing of the input application. Long-term behavior in a system is predicted using LTI systems. The term “linear translation-invariant” can be used to describe these systems, giving it the broadest meaning possible. The analogous term in the case of generic discrete-time (i.e., sampled) systems is linear shift-invariant.
Table of Content
- LTI System
- Types
- Properties
- Transfer Function
- Convolution
- Sampling Theorem
- Nyquist Rate