Zak Transform
Yes! There is a transform called the Zak Transform, though it’s much less commonly mentioned than the Fourier or Z-transform. It comes up mostly in signal processing, time-frequency analysis, and communications.
1. Definition
The Zak Transform of a signal x(t) (continuous-time) is defined as:
$$
Z_x(t, f) = \sum_{n=-\infty}^{\infty} x(t - nT) e^{j 2 \pi n f T}
$$
Where:
x(t)is the input signalTis a fixed periodtis a continuous variable within [0, T)fis the frequency variable
It essentially maps a 1D signal into a 2D representation in time-frequency space.
2. Key Properties
- Periodicity: $ Z_x(t + T, f) = e^{j 2 \pi f T} Z_x(t, f) $
-
Time-Frequency Analysis:
- Converts time-domain signal into a representation showing both time and frequency structure.
- Often used in Gabor frames, multicarrier communications, and OFDM.
-
Invertible: There’s an inverse Zak Transform to recover
x(t).
4. Discrete Zak Transform
$$
Z_x[m, k] = \sum_{n=0}^{N-1} x[n] e^{-j 2\pi kn / N} \delta_{n \mod M, m}
$$
Where N is the signal length and M is a chosen sub-sampling period.
Step A: Receive the Signal
$$
y(t) = x(t) * h(t) + w(t)
$$
Step B: Apply the Zak Transform
$$
Z_y(t, f) = \sum_{n} y(t - nT) e^{j 2 \pi n f T}
$$
$$
Z_y(t,f) = Z_x(t,f) \cdot H_{\text{eff}}(t,f) + Z_w(t,f)
$$
Step C: Equalization
$$
\hat{Z}_x(t,f) = Z_y(t,f) / H_{\text{eff}}(t,f)
$$
Step D: Inverse Zak Transform
$$
x(t) = \int_0^T Z_x(t,f) df
$$