Time Reversal in the Frequency Domain & TRM

Time Reversal Mirror (TRM) and Fourier Transform

1. Time Reversal and Convolution

In TRM, the received signal at a transducer is:

y_n(t) = x(t) * h_n(t)

Here, x(t) is the transmitted signal and h_n(t) is the channel impulse response. After time reversal, the signal becomes:

y_n^*(-t) = x^*(-t) * h_n^*(-t)

Key property: Convolution in the time domain corresponds to multiplication in the frequency domain:

ℱ{x(t) * h(t)} = X(f) H(f)

So in the frequency domain, the TRM operation becomes:

Z(f) = X^*(f) ∑_n |H_n(f)|^2

2. Time Reversal in the Frequency Domain

Time reversal in time domain (t → -t) corresponds to complex conjugation in the frequency domain:

ℱ{x^*(-t)} = X^*(f)

• This is exactly why, after retransmission, the combined signal effectively becomes:

  z(t) ≈ x^*(-t)

• The sum (∑_n |H_n(f)|^2) acts like a frequency-dependent gain, and if energy is concentrated, it is nearly constant, so the original signal is “focused” back at the source.

3. Energy Focusing Property

This is the Fourier-related property that TRM exploits:

• The autocorrelation property of a signal: Convolution of a function with its time-reversed conjugate is equivalent to autocorrelation in the time domain. In the frequency domain, this is multiplication by the magnitude squared:

  h(t) * h^*(-t) ↔ |H(f)|^2

• This explains why TRM focuses energy in both space and time—it essentially concentrates the channel energy toward the original source.

Summary

1. Convolution-Multiplication Duality: Convolution in time ↔ Multiplication in frequency.
2. Time Reversal ↔ Complex Conjugation: x^*(-t) ↔ X^*(f).
3. Autocorrelation: h(t) * h^*(-t) ↔ |H(f)|^2.

These properties make TRM very powerful in underwater acoustic communication, because even in complex multipath channels, the time-reversed retransmission naturally refocuses energy.