Signals and System Transforms

Signals & System Transforms

List of Transforms in Signals and Systems

1. Fourier Transform

• Converts a time-domain signal to its frequency-domain representation.
• Continuous-Time Fourier Transform (CTFT):
  $X(j\omega )={\int }_{-\infty }^{\infty }x(t)e^{-j\omega t}dt$
• Discrete-Time Fourier Transform (DTFT):
  $X(e^{j\omega })={\sum }_{n=-\infty }^{\infty }x[n]e^{-j\omega n}$

2. Inverse Fourier Transform

• Recovers the time-domain signal from its frequency-domain representation.
• CTFT Inverse:
  $x(t)=\frac{1}{2\pi }{\int }_{-\infty }^{\infty }X(j\omega )e^{j\omega t}d\omega$
• DTFT Inverse:
  $x[n]=\frac{1}{2\pi }{\int }_{-\pi }^{\pi }X(e^{j\omega })e^{j\omega n}d\omega$

3. Laplace Transform

• Analyzes signals in the $s$-domain (complex frequency domain).
• Definition:
  $X(s)={\int }_0^{\infty }x(t)e^{-st}dt$

4. Inverse Laplace Transform

• Converts back to the time-domain from the $s$-domain.
• Definition:
  $x(t)=\frac{1}{2\pi j}{\int }_{\sigma -j\infty }^{\sigma +j\infty }X(s)e^{st}ds$

5. Z-Transform

• Discrete counterpart of the Laplace Transform, useful for discrete signals.
• Definition:
  $X(z)={\sum }_{n=-\infty }^{\infty }x[n]z^{-n}$

6. Inverse Z-Transform

• Recovers the discrete-time signal from the $z$-domain.
• Definition:
  $x[n]=\frac{1}{2\pi j}\oint X(z)z^{n-1}dz$

7. Discrete Fourier Transform (DFT)

• Frequency representation of discrete-time finite-duration signals.
• Definition:
  $X[k]={\sum }_{n=0}^{N-1}x[n]e^{-j\frac{2\pi }{N}kn}$

8. Inverse Discrete Fourier Transform (IDFT)

• Converts back from frequency-domain to discrete-time domain.
• Definition:
  $x[n]=\frac{1}{N}{\sum }_{k=0}^{N-1}X[k]e^{j\frac{2\pi }{N}kn}$

9. Fast Fourier Transform (FFT)

• Algorithm to compute the DFT efficiently.

10. Hilbert Transform

• Generates the analytic signal by introducing a phase shift of $90^{\circ }$.
• Definition:
  $\hat{x}(t)=\frac{1}{\pi }{\int }_{-\infty }^{\infty }\frac{x(\tau )}{t-\tau }d\tau$

11. Wavelet Transform

• Analyzes signals in both time and frequency domains simultaneously.
• Continuous Wavelet Transform (CWT):
  $W(a,b)={\int }_{-\infty }^{\infty }x(t){\psi }_{a,b}^{\ast }(t)dt$
• Discrete Wavelet Transform (DWT): Computed via filters.

12. Short-Time Fourier Transform (STFT)

• Analyzes signals in small time segments for time-frequency representation.
• Definition:
  $X(t,\omega )={\int }_{-\infty }^{\infty }x(\tau )w(\tau -t)e^{-j\omega \tau }d\tau$

13. Hadamard Transform

• Used for signal compression and data processing.
• Recursive definition using Walsh functions.

14. Cosine Transform

• Variant of Fourier Transform using cosine basis.
• Discrete Cosine Transform (DCT):
  $X[k]={\sum }_{n=0}^{N-1}x[n]\cos \left[\frac{\pi }{N}\left(n+\frac{1}{2}\right)k\right]$

15. Radon Transform

• Projects a 2D function onto a 1D space.

16. Chirp Z-Transform

• Computes Z-transform over a specified contour in the $z$-domain.

17. Hartley Transform

• Alternative to Fourier Transform using cosine and sine simultaneously.
• Definition:
  $H(\omega )={\int }_{-\infty }^{\infty }x(t)[\cos (\omega t)+\sin (\omega t)]dt$

18. S-Transform

• Time-frequency localization similar to Wavelet Transform.

References

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References

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• date: 2024.11.25
• time: 13:37>)