Invertible Systems

Introduction

A system is invertible if distinct inputs always produce distinct outputs — meaning you can uniquely determine what went in by observing what came out. Invertibility is essential in any application where you need to recover the original signal from the system's output: equalization in communication channels, deblurring in image processing, and deconvolution in seismology all require the underlying system to be invertible.

Think of it like encoding a message. If two different messages can produce the same encoded output, the receiver cannot determine which message was sent. An invertible system guarantees a one-to-one mapping between inputs and outputs, making perfect recovery of the input theoretically possible.

Formal Definition

A system $T$ is invertible if and only if distinct inputs produce distinct outputs:

$$x_1(t) \neq x_2(t) \implies T\{x_1(t)\} \neq T\{x_2(t)\}$$

Equivalently, the mapping from input to output is one-to-one (injective). When a system is invertible, there exists an inverse system $T^{-1}$ such that:

$$T^{-1}\{T\{x(t)\}\} = x(t) \quad \text{for all valid inputs } x(t)$$

The cascade of the system followed by its inverse recovers the original input.

Examples of Invertible Systems

Example 1: Amplitude Scaling

$y(t) = cx(t)$ where $c \neq 0$.

Inverse: $x(t) = (1/c) \cdot y(t)$. This is invertible for any non-zero $c$. If $c = 0$, every input maps to zero output — not invertible.

Example 2: Time Delay

$y(t) = x(t - t_0)$.

Inverse: $x(t) = y(t + t_0)$. A delay is perfectly invertible by advancing the output.

Example 3: Accumulator (Discrete-Time)

$y[n] = \sum_{k=-\infty}^{n} x[k]$

Inverse: $x[n] = y[n] - y[n-1]$ (first difference). The running sum is invertible because differencing recovers the original sequence.

Example 4: Differentiation

$y(t) = dx(t)/dt$

Inverse: $x(t) = \int_{-\infty}^{t} y(\tau) d\tau + x(-\infty)$. This is invertible only if the initial condition $x(-\infty)$ is known; otherwise, the constant component is lost.

Examples of Non-Invertible Systems

Example 1: Squaring

$y(t) = x^2(t)$.

Both $x(t) = A$ and $x(t) = -A$ produce the same output $y(t) = A^2$. The sign information is permanently lost — not invertible.

Example 2: Full-Wave Rectification

$y(t) = |x(t)|$.

Similar to squaring: $+A$ and $-A$ both map to $A$. Not invertible.

Example 3: Sampling Below Nyquist

When a signal is sampled below the Nyquist rate, multiple different continuous signals can produce the same set of samples (due to aliasing). The sampling system becomes non-invertible.

Example 4: Quantization

$y = Q(x)$ maps a range of values to a single level. Many different inputs map to the same output — not invertible. This is why quantization is inherently lossy.

Conditions for Invertibility of LTI Systems

For Linear Time-Invariant systems, invertibility has elegant characterizations:

Continuous-Time LTI Systems

An LTI system with impulse response $h(t)$ is invertible if and only if there exists $h_i(t)$ such that: $$h(t) * h_i(t) = \delta(t)$$

In the frequency domain: $H(j\omega) \cdot H_i(j\omega) = 1$, giving: $$H_i(j\omega) = \frac{1}{H(j\omega)}$$

The inverse system has transfer function equal to the reciprocal of the original. This is possible if and only if $H(j\omega) \neq 0$ for all $\omega$ — the system must not have any zeros on the frequency axis.

Discrete-Time LTI Systems

Similarly, $H(z) \cdot H_i(z) = 1$ requires: $$H_i(z) = \frac{1}{H(z)}$$

The zeros of $H(z)$ become the poles of $H_i(z)$. For the inverse to be causal and stable, all zeros of $H(z)$ must be inside the unit circle (the system must be minimum phase).

Minimum Phase and Invertibility

A system is minimum phase if all its zeros lie inside the unit circle (discrete-time) or in the left half-plane (continuous-time). Minimum-phase systems have causal, stable inverses. Non-minimum-phase systems have inverses that are either non-causal or unstable, making practical inversion difficult.

Example: $H(z) = 1 - 2z^{-1}$ has a zero at $z = 2$ (outside unit circle). Its inverse $H_i(z) = \frac{1}{1-2z^{-1}}$ has a pole at $z = 2$ — unstable if causal. This system is non-minimum phase and cannot be practically inverted with a causal stable filter.

Applications of Inverse Systems

Channel equalization: In communications, the channel distorts the transmitted signal: $y = h * x$. An equalizer implements $h_i$ to recover $x$: $h_i * y = h_i * h * x = x$.

Deconvolution: In seismology, the Earth's response $h(t)$ smears the source wavelet. Deconvolution applies the inverse filter to sharpen reflections.

Image deblurring: Optical blur is modeled as convolution with a point spread function. The inverse restores sharpness (though noise limits practical performance).

Audio de-reverberation: Room acoustics convolve dry speech with a room impulse response. Inverse filtering can, in principle, remove reverberation.

Practical Limitations

Even when a system is theoretically invertible, practical inversion faces challenges:

• Noise amplification: Where $|H(j\omega)|$ is small, $|1/H(j\omega)|$ is large, amplifying noise
• Stability: Non-minimum-phase systems require non-causal or unstable inverses
• Model mismatch: If $H$ is not perfectly known, the inverse is imperfect

Regularization techniques (Wiener filtering, Tikhonov regularization) address these by trading perfect inversion for robustness against noise.

Key Takeaways

• A system is invertible if distinct inputs always produce distinct outputs (one-to-one mapping)
• The inverse system satisfies $T^{-1}\{T\{x\}\} = x$ for all inputs
• LTI invertibility requires $H(j\omega) \neq 0$ for all frequencies — no spectral nulls
• The inverse transfer function is $H_i = 1/H$ — zeros become poles
• Minimum-phase systems have causal, stable inverses; non-minimum-phase systems do not
• Applications include equalization, deconvolution, deblurring, and de-reverberation
• Noise amplification is the practical limit on inverse filtering performance