Causality And Stability Of Discrete Time Systems

Discrete Time Systems

Discrete time systems A discrete-time system is a system that maps an input sequence with an output sequence

y[n] = T {x[n]}

A discrete-time system is a device or algorithm that, according to some well-defined rule, operates on a discrete-time signal called the input signal or excitation to produce another discrete-time signal called the output signal or response. Mathematically speaking, a system is also a function. The input signal x[n] is transformed by the system into a signal y[n], which we express mathematically as

y[·] = T {x[·]} or y[n] = T {x[·]}[n] or x[·] T→ y[·]

The notation y[n] = T {x[n]} is mathematically vague. The reader must understand that in general y[n] is a function of the entire sequence {x[n]}, not just the single time point x[n].

Input Signal x[n]→ Discrete-time system → Output Signal y[n]

Causality

If a discrete-time system is causal, then the current output sample depends only on the current and past input samples and not on future samples. This definition is now examined in the case of an LSI system with an impulse response h[n]. Therefore, an LSI system is causal if and only if h[n] = 0, n < 0. Fora causal system, the output y[n] at any time n depends only on the “present” and “past” inputs i.e.,

y[n] = F{x[n], x[n − 1], x[n − 2], . . .}

where F{·} is any function.

Causality is necessary for real-time implementation, but many DSP problems involved stored data, e.g., image processing (OCR) or restoration of analog audio recordings. Otherwise noncausal system.

Stability

BIBO stability stands for bounded input, bounded output stability. BIBO stability is the system property that any bounded input yields a bounded output. This is to say that as long as we input a signal with absolute value less than some constant, we are guaranteed to have an output with absolute value less than some other constant. A system is bounded-input bounded-output (BIBO) stable iff every bounded input produces a bounded output.

∃Mx s.t. |x[n]| ≤ Mx < ∞ ∀n, then there must exist an My s.t. |y[n]| ≤ My < ∞ ∀n.

Usually My will depend on Mx.

Example:-

accumulator y[n] = y[n − 1] + x[n]. Consider input signal x[n] = u[n], which is bounded by Mx = 1. But y[n] = n + 1 + y[−1] blows up, so the accumulator is an unstable system. We will derive a simple test for BIBO stability shortly.