Impulse Response & LTI Systems | DSP-101

h[n]: The Complete Portrait of an LTI System

In the previous lesson we derived the convolution sum and showed that the output of any LTI system is y[n] = x[n] * h[n]. That h[n] — the impulse response — is remarkable: a single sequence that completely and uniquely characterizes the behavior of the entire system. Feed any input into an LTI system, convolve it with h[n], and you get the exact output. No other parameters are needed.

The impulse response is measured by applying the unit impulse δ[n] as the input. Because δ[n] is the convolution identity — x[n] * δ[n] = x[n] for any x[n] — the output when x[n] = δ[n] is exactly h[n] itself. This gives engineers a practical measurement technique: excite the system with a brief, broadband pulse and observe the response.

Definition of Impulse Response

h[n] = T\{\delta[n]\}

When the input is the unit impulse δ[n], the output is the impulse response h[n]. This single measurement fully characterizes the LTI system for all possible inputs.

Why is h[n] a complete characterization? Because linearity and time-invariance together guarantee that the response to any input can be computed as a superposition of scaled, shifted copies of h[n]. If you know how the system responds to a single impulse, you know how it responds to everything.

FIR Systems: Finite Impulse Response

An LTI system is called FIR (Finite Impulse Response) when h[n] is nonzero for only a finite number of samples — say, for n = 0, 1, …, M−1 and zero everywhere else. The convolution sum then reduces to a finite weighted sum of M past and present input samples:

FIR Output Equation

y[n] = \sum_{k=0}^{M-1} h[k]\,x[n-k]

The output is a weighted sum of the current and M−1 past input samples. The weights h[0], h[1], …, h[M−1] are the filter coefficients — identical to the impulse response values.

FIR systems have two outstanding properties. First, they are always stable: if the input is bounded (|x[n]| ≤ B for all n), then the output is bounded by B times the sum of |h[k]| over the finite support, which is always finite. Second, FIR systems can achieve exactly linear phase — all frequencies are delayed by the same number of samples — simply by making h[n] symmetric. Linear phase means no phase distortion, a requirement in audio, instrumentation, and communications equalization.

Property 1

Always Stable

Finite h[n] ⇒ finite sum ⇒ bounded output for any bounded input. No stability analysis required.

Property 2

Linear Phase

Symmetric coefficients h[n] = h[M−1−n] guarantee all frequencies experience the same group delay.

Property 3

No Feedback

Output depends only on present and past inputs — no recursive loops. Simple and robust implementation.

Property 4

Higher Cost

Sharp frequency selectivity requires many taps M. FFT-based fast convolution mitigates the cost for large M.

IIR Systems: Infinite Impulse Response

An LTI system is IIR (Infinite Impulse Response) when h[n] extends infinitely — it never reaches zero in finite time. The exponential sequence h[n] = αⁿu[n] is a classic example: for |α| < 1 it decays toward zero but theoretically never vanishes.

Computing an infinite convolution sum directly is not practical. Instead, IIR systems are implemented via difference equations that express the current output as a linear combination of past outputs and present/past inputs. The feedback through past outputs is what generates the infinite impulse response:

General IIR Difference Equation

y[n] = \sum_{k=0}^{N} b_k\,x[n-k] - \sum_{k=1}^{M} a_k\,y[n-k]

The b_k coefficients weight the inputs (feedforward path); the a_k coefficients weight the past outputs (feedback path). This recursive structure generates an impulse response that can extend indefinitely.

The efficiency advantage of IIR filters is significant: an IIR filter can achieve the same frequency-selectivity as an FIR filter with far fewer coefficients. A 4th-order Butterworth IIR low-pass filter, for example, can match what might require 50–200 FIR taps. The trade-off is that IIR systems require explicit stability analysis (all poles inside the unit circle in the Z-domain), and they cannot achieve exactly linear phase in general.

First-Order IIR Example

Consider the simplest IIR filter: y[n] = α·y[n−1] + x[n]. When x[n] = δ[n], the output is:

y[0] = α·y[−1] + δ[0] = 0 + 1 = 1

y[1] = α·y[0] + δ[1] = α + 0 = α

y[2] = α·y[1] = α², and so on.

h[n] = αⁿu[n] — the impulse response is infinite but decays when |α| < 1.

Difference Equations as System Descriptions

A linear constant-coefficient difference equation (LCCDE) is the time-domain description of an LTI system in the same way that a differential equation describes a continuous-time system. Every digital filter designed in practice — from audio equalizers to control-system compensators — is ultimately described by an LCCDE of some order.

The order of the system equals the maximum delay in the recursion: a system that uses y[n−1] and y[n−2] is second-order. Higher order means more poles, which allows sharper frequency transitions but demands more careful stability monitoring. The coefficients {a_k, b_k} are the design parameters; changing them changes the filter's frequency response.

Second-Order Section (Biquad)

y[n] = b_0 x[n] + b_1 x[n-1] + b_2 x[n-2] - a_1 y[n-1] - a_2 y[n-2]

The biquad (second-order section) is the workhorse building block. Any higher-order IIR filter is implemented as a cascade of biquads to maximize numerical stability.

Even an FIR filter has an LCCDE — it just has no feedback terms (all a_k = 0 except a₀ = 1). The difference equation then reduces to the finite convolution sum. So LCCDEs are a unified description: set all feedback coefficients to zero and you have an FIR; include feedback and you have an IIR.

Cascade Combinations

When two LTI systems are connected in cascade (the output of the first feeds the input of the second), the combined system is itself LTI. By the associative property of convolution, the overall impulse response is the convolution of the individual impulse responses:

Cascade Combination

h[n] = h_1[n] * h_2[n]

Two cascaded LTI systems with impulse responses h₁[n] and h₂[n] behave as a single system with impulse response h[n] = h₁[n] * h₂[n].

Three important consequences follow. First, the order of cascade does not matter: applying h₁ then h₂ produces the same result as h₂ then h₁, because convolution is commutative. Second, a bank of cascaded filters can be pre-combined offline into a single impulse response, reducing real-time computation. Third, a cascade of stable LTI systems is stable as long as each individual system is stable.

Parallel Combinations

When two LTI systems receive the same input and their outputs are added, they form a parallel combination. By the distributive property of convolution, the overall impulse response is simply the sum of the individual impulse responses:

Parallel Combination

h[n] = h_1[n] + h_2[n]

Two parallel LTI systems with impulse responses h₁[n] and h₂[n] behave as a single system with h[n] = h₁[n] + h₂[n].

Parallel structures are everywhere in practice. A graphic equalizer is a bank of bandpass filters (each an LTI system) whose outputs are added after independent gain scaling. An OFDM receiver processes multiple subcarriers simultaneously, each going through its own matched filter. In all these cases, the combined system is still LTI and its total impulse response is the sum of the branches.

Stability Criterion for IIR Systems

An IIR system described by an LCCDE is BIBO stable if and only if its impulse response is absolutely summable:

Σ_n=−∞^∞ |h[n]| < ∞

In the Z-transform domain, this condition is equivalent to requiring that all poles of the transfer function H(z) lie strictly inside the unit circle. For the first-order IIR y[n] = αy[n−1] + x[n], stability requires |α| < 1.

The next module introduces the Z-Transform — a powerful algebraic tool that converts difference equations into polynomial equations in the complex variable z, making system analysis, stability checking, and filter design far more tractable.

Key Takeaways

The impulse response h[n] completely characterizes an LTI system: knowing h[n] is sufficient to compute the output for any input via convolution y[n] = x[n] * h[n].
FIR systems have finite-length h[n], always stable, and can achieve exactly linear phase by using symmetric coefficients.
IIR systems have infinite-length h[n] implemented via recursive difference equations; they achieve sharp selectivity with few coefficients but require stability analysis.
A general LCCDE y[n] = Σb_k·x[n−k] − Σa_k·y[n−k] describes any LTI filter; the second-order section (biquad) is the standard building block for IIR designs.
Cascaded LTI systems have a combined impulse response h₁[n] * h₂[n]; the order of cascade is interchangeable.
Parallel LTI systems have a combined impulse response h₁[n] + h₂[n].
An IIR system is BIBO stable if and only if Σ|h[n]| < ∞, equivalently, all poles of H(z) are inside the unit circle.