Time Reversal

Introduction

Time reversal means playing a signal backward — what happened last comes first, and what happened first comes last. Mathematically, it replaces every time index $t$ with $-t$, effectively reflecting the signal about the vertical axis. While this might seem like a purely academic manipulation, time reversal is deeply embedded in practical signal processing: it is fundamental to correlation operations, matched filter design, the even-odd decomposition, and even advanced techniques in medical ultrasound imaging.

When you compute the cross-correlation between two signals, you are actually convolving one signal with the time-reversed version of the other. When you design a matched filter (the optimal detector for a known signal in noise), its impulse response is the time-reversed version of the signal you want to detect. Understanding time reversal thoroughly unlocks your ability to work with these important concepts.

Mathematical Definition

Continuous-Time

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

Every value that was at time $t$ in the original signal appears at time $-t$ in the reversed signal.

Discrete-Time

$$y[n] = x[-n]$$

The sequence is reflected about the origin $n = 0$.

Combined with Time Shift

When time reversal is combined with a time shift, the order of operations matters:

• Reverse then shift by $t_0$: $y(t) = x(-(t - t_0)) = x(-t + t_0)$ — the reversed signal shifts right by $t_0$
• Shift then reverse: $y(t) = x(-(t) - t_0) = x(-t - t_0)$ — different result

In graphical problems, always apply operations in the specified order.

Graphical Interpretation

Consider a signal $x(t)$ that starts at $t = 1$, peaks at $t = 3$, and ends at $t = 5$. After time reversal:

• The start point moves to $t = -1$
• The peak moves to $t = -3$
• The end point moves to $t = -5$
• The signal now occupies $-5 \leq t \leq -1$

The signal appears as a mirror image about the vertical axis ($t = 0$). Every feature that was to the right of the origin is now equidistant to the left.

Effect on Transforms

Fourier Transform

For a real-valued signal: $$\mathcal{F}\{x(-t)\} = X(-j\omega) = X^*(j\omega)$$

Time reversal conjugates the spectrum (for real signals). The magnitude spectrum $|X(j\omega)|$ remains unchanged (since $|X^*| = |X|$), but the phase spectrum flips sign: $\angle X(-j\omega) = -\angle X(j\omega)$.

For complex signals, time reversal gives $X(-j\omega)$ without conjugation.

Laplace Transform

$$\mathcal{L}\{x(-t)\} = X(-s)$$

The ROC also reflects about the imaginary axis. If the original ROC was $\text{Re}(s) > \sigma_0$, the time-reversed signal's ROC becomes $\text{Re}(s) < -\sigma_0$.

Z-Transform

$$\mathcal{Z}\{x[-n]\} = X(z^{-1})$$

Replace $z$ with $1/z$. The ROC inverts: if it was $|z| > r$, it becomes $|z| < 1/r$.

Properties of Time Reversal

Double reversal (involution): Reversing twice gives back the original: $$x(-(-t)) = x(t)$$

Energy preservation: Time reversal does not change signal energy: $$\int_{-\infty}^{\infty}|x(-t)|^2 dt = \int_{-\infty}^{\infty}|x(t)|^2 dt = E_x$$

The substitution $\tau = -t$ proves this immediately.

Even signals are invariant: If $x(-t) = x(t)$, the signal is even (symmetric about $t = 0$). Time reversal leaves it unchanged.

Odd signals negate: If $x(-t) = -x(t)$, the signal is odd (anti-symmetric). Time reversal is equivalent to negation.

Even-Odd Decomposition

Every signal can be uniquely decomposed into even and odd components using time reversal:

$$x_e(t) = \frac{x(t) + x(-t)}{2} \qquad (\text{even part})$$

$$x_o(t) = \frac{x(t) - x(-t)}{2} \qquad (\text{odd part})$$

The original signal is: $x(t) = x_e(t) + x_o(t)$

This decomposition is important because:

• The Fourier Transform of the even part gives the real part of the spectrum
• The Fourier Transform of the odd part gives the imaginary part of the spectrum
• Even signals have purely real spectra; odd signals have purely imaginary spectra

Example

Let $x(t) = e^{-t}u(t)$ (causal decaying exponential).

Even part: $x_e(t) = \frac{e^{-t}u(t) + e^{t}u(-t)}{2} = \frac{1}{2}e^{-|t|}$

Odd part: $x_o(t) = \frac{e^{-t}u(t) - e^{t}u(-t)}{2} = \frac{1}{2}\text{sgn}(t) \cdot e^{-|t|}$

Applications

Matched Filtering

The matched filter is the optimal linear filter for detecting a known signal $s(t)$ embedded in white Gaussian noise. Its impulse response is:

$$h(t) = s(T - t) = s(-(t - T))$$

This is the time-reversed (and delayed) version of the expected signal. The output is maximized at $t = T$, giving the maximum signal-to-noise ratio. Matched filters are used in radar (detecting returning pulses), digital communications (detecting transmitted symbols), and sonar.

Correlation via Convolution

Cross-correlation between $x(t)$ and $h(t)$ is defined as: $$R_{xh}(\tau) = \int_{-\infty}^{\infty} x(t) h(t + \tau) dt$$

This can be written as convolution with a time-reversed signal: $$R_{xh}(\tau) = x(\tau) * h(-\tau)$$

This relationship is computationally powerful: correlation can be computed using the same efficient algorithms (FFT-based) as convolution, simply by time-reversing one signal first.

Autocorrelation

The autocorrelation $R_{xx}(\tau) = x(\tau) * x(-\tau)$ is always an even function (symmetric about $\tau = 0$) with its maximum at the origin. The Fourier Transform of the autocorrelation gives the power spectral density (Wiener-Khinchin theorem).

Time-Reversal Acoustics

In medical ultrasound and underwater acoustics, time-reversal techniques exploit the fact that wave equations are symmetric in time. A pulse scattered by a target is recorded, time-reversed, and retransmitted. The retransmitted pulse naturally refocuses on the original scattering point, even through complex media. This enables focusing through inhomogeneous tissue without knowing the medium's properties.

Worked Example

Problem: Given $x[n] = \{1, 2, 3, 4, 5\}$ for $n = 0, 1, 2, 3, 4$, find $x[-n]$.

Solution: Replace each index $n$ with $-n$:

• $x[0] = 1$ stays at $n = 0$
• $x[-1] = x[1] = 2$ appears at $n = -1$
• $x[-2] = x[2] = 3$ appears at $n = -2$
• $x[-3] = x[3] = 4$ appears at $n = -3$
• $x[-4] = x[4] = 5$ appears at $n = -4$

Result: $y[n] = x[-n] = \{5, 4, 3, 2, 1\}$ for $n = -4, -3, -2, -1, 0$

The sequence is mirrored about $n = 0$.

Key Takeaways

• Time reversal: $x(t) \to x(-t)$ reflects the signal about $t = 0$
• For real signals: $\mathcal{F}\{x(-t)\} = X^*(j\omega)$ — magnitude preserved, phase negated
• Even signals are unchanged by reversal; odd signals change sign
• Enables the even-odd decomposition: $x_e = (x(t) + x(-t))/2$
• Cross-correlation equals convolution with a time-reversed signal
• Matched filter impulse response is the time-reversed signal to be detected
• Time reversal preserves signal energy