LTI Systems
LTI Systems and Convolution
An LTI system is a system that is both linear (obeys the superposition principle) and time-invariant (its behavior doesn't change over time).
The output y(t) of an LTI system is determined by the convolution of the input signal x(t) with the system's unique impulse response h(t). Convolution is a mathematical operation expressed as:
y(t) = ∫-∞∞ x(Ï„) h(t - Ï„) dÏ„
1. For a Discrete-Time System
In wireless communication, the signal reaches the receiver thru different multi-paths. And they are nothing but time-delayed versions of the same transmitted signal. You always find a relationship between received and transmitted signals is
y = h * x + n
which denotes that the transmitted signal 'x' is convolved with the channel coefficients ('h') of the particular channel. Because here time-shifted version of the initially transmitted signal is overlapped with the channel coefficients because the time-delayed version of 'x' reaches the receiver at a different time or creates a delay. So, it is necessary to consider all delayed versions of the same signal for a better approximation of transmitted symbols or bits.
We receive multiple impulse responses for a particular input signal delta or unit impulse transmission.
i.e., the particular input
x[n] = δ[n]
Produces the output
y[n] = h[n]
**h[n] = ..., h[-2], h[-1], h[0], h[1], h[2], .... (impulse responses due to multipath etc.,)
So the general input is going to be
x[n] = x[k]δ[n-k] (on an interval of -infinity to +infinity)
will thus produce the output
y[n] = x[k]h[n-k] (on an interval of -infinity to +infinity)
Which is also termed a 'convolution sum.'
2. For a Continuous-Time System
For a typical wireless communication system, x is the transmitted data signal, and h is the channel impulse response. And their convolution is represented by this.
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The LTI system is modeled considering the original signal convolved with the channel impulse response. On the other hand, on the receiver side, the signal is retrieved by using equalizers. That estimates the originally transmitted signal from the training bits / symbols.