Signals And Systems Question Paper 1

Laplace Transform: The Laplace transform $X (s)$ of a continuous-time signal $x (t)$ is given by:
$X (s) = \int_{0}^{\infty} x (t) e^{- s t} d t$
where $s$ is a complex variable: $s = σ + jω$ .
Inverse Laplace Transform: The inverse Laplace transform is used to find $x (t)$ from $X (s)$ and can be computed using:
$x (t) = L^{- 1} {X (s)}$
This can be computed using partial fraction expansion or the residue theorem, depending on the problem.
Fourier Transform: The Fourier Transform of a continuous-time signal $x (t)$ is given by:
$X (f) = \int_{- \infty}^{\infty} x (t) e^{- j 2 π f t} d t$
where $f$ is the frequency variable.
Inverse Fourier Transform: The inverse Fourier Transform is used to recover $x (t)$ from $X (f)$ and is given by:
$x (t) = \int_{- \infty}^{\infty} X (f) e^{j 2 π f t} df$
Convolution of Continuous-Time Signals: The convolution $y (t) = (x * h) (t)$ of two continuous-time signals $x (t)$ and $h (t)$ is defined as:
$y (t) = \int_{- \infty}^{\infty} x (τ) h (t - τ) d τ$

Z-Transform: The Z-transform $X (z)$ of a discrete-time signal $x [n]$ is given by:
$X (z) = n = - \infty \sum \infty x [n] z^{- n}$
where $z$ is a complex variable.
Inverse Z-Transform: The inverse Z-transform is used to recover $x [n]$ from $X (z)$ and is generally computed using methods like partial fraction expansion or long division.
Fourier Transform (Discrete-Time): The Discrete-Time Fourier Transform (DTFT) of a discrete-time signal $x [n]$ is given by:
$X (f) = n = - \infty \sum \infty x [n] e^{- j 2 π f n}$
where $f$ is the frequency variable.
Inverse Fourier Transform (Discrete-Time): The inverse DTFT is used to recover $x [n]$ from $X (f)$ and is given by:
$x [n] = \int_{- \infty}^{\infty} X (f) e^{j 2 π f n} df$
Convolution of Discrete-Time Signals: The convolution $y [n] = (x * h) [n]$ of two discrete-time signals $x [n]$ and $h [n]$ is defined as:

y [n] = m = - \infty \sum \infty x [m] h [n - m]

The Laplace transform is for continuous-time signals, while the Z-transform is for discrete-time signals.
For both continuous and discrete signals, there are corresponding Fourier transforms (continuous-time Fourier transform and discrete-time Fourier transform), but the DTFT uses summation over discrete indices, while the CTFT uses integration over continuous time.

References