Unit 2: Transforms

Introduction to Fourier and Z-Transforms

Transforms are mathematical tools used to convert a function from one domain to another. In this unit, we will cover two important transforms: the Fourier Transform (FT) and the Z-Transform (ZT). These transforms have wide applications in signal processing, control systems, image processing, and more.

Fourier Transform (FT)

The Fourier Transform helps in analyzing functions by decomposing them into sine and cosine waves. It is particularly useful in studying signals and systems in the frequency domain.

Complex Exponential Form of Fourier Series

The Fourier series represents a periodic function as a sum of sines and cosines. When expressed in complex exponential form, the series becomes:

$f(x) = \sum\_{n=-\infty}^{\infty} C_n e^{i n \omega_0 x}$

where $C_n$ are the Fourier coefficients, and $\omega_0$ is the fundamental frequency.

Example:

For a square wave with period $2\pi$, the Fourier series is:

$f(x) = \frac{4}{\pi} \sum\_{n=1,3,5...}^{\infty} \frac{1}{n} \sin(nx)$

Fourier Integral Theorem

The Fourier Integral Theorem states that a function can be represented by an integral of sinusoidal components if it is not periodic but decays to zero at infinity.

Example:

For a function $f(t)$ that decays at infinity, its Fourier integral is:

$f(t) = \int\_{-\infty}^{\infty} F(\omega) e^{i \omega t} d\omega$

where $F(\omega)$ is the Fourier transform of $f(t)$.

Fourier Sine and Cosine Integrals

The Fourier Sine and Cosine Integrals decompose a function into only sine or cosine terms. These are useful when dealing with odd or even functions.

Example:

For an odd function $f(t)$, the Fourier sine integral is:

$f(t) = \int_0^{\infty} F_s(\omega) \sin(\omega t) d\omega$

Fourier Transform (FT)

The Fourier Transform converts a time-domain signal into its frequency-domain representation:

$F(\omega) = \int\_{-\infty}^{\infty} f(t) e^{-i \omega t} dt$

Example:

The Fourier transform of a Gaussian function $f(t) = e^{-t^2}$ is:

$F(\omega) = \sqrt{\pi} e^{-\frac{\omega^2}{4}}$

Fourier Sine and Cosine Transforms

These transforms use sine and cosine functions rather than exponentials.

• Fourier Sine Transform (FST):

$F_s(\omega) = \int_0^{\infty} f(t) \sin(\omega t) dt$

• Fourier Cosine Transform (FCT):

$F_c(\omega) = \int_0^{\infty} f(t) \cos(\omega t) dt$

Example:

For $f(t) = e^{-t}$, the Fourier sine transform is:

$F_s(\omega) = \frac{\omega}{1 + \omega^2}$

Inverse Fourier Transform

To recover the time-domain signal from its frequency-domain form, the inverse Fourier transform is used:

$f(t) = \frac{1}{2\pi} \int\_{-\infty}^{\infty} F(\omega) e^{i \omega t} d\omega$

Discrete Fourier Transform (DFT)

The Discrete Fourier Transform (DFT) is used when dealing with discrete data points. It converts a sequence of values into components of different frequencies.

Definition

For a sequence $x[n]$ of length $N$, the DFT is given by:

$X[k] = \sum\_{n=0}^{N-1} x[n] e^{-i \frac{2\pi}{N} kn}$

Example:

For a sequence $x[n] = [1, 2, 3, 4]$, the DFT computes the frequency components of this sequence.

Z-Transform (ZT)

The Z-Transform is a powerful tool for analyzing discrete-time systems. It is the discrete counterpart of the Laplace transform and is widely used in digital signal processing.

Introduction and Definition

The Z-transform of a discrete sequence $x[n]$ is defined as:

$X(z) = \sum\_{n=0}^{\infty} x[n] z^{-n}$

where $z$ is a complex number.

Standard Properties of Z-Transform

1. Linearity: If $X(z)$ and $Y(z)$ are the Z-transforms of $x[n]$ and $y[n]$, respectively, then:

   $aX(z) + bY(z) \text{ is the Z-transform of } ax[n] + by[n]$

2. Time Shifting: If $X(z)$ is the Z-transform of $x[n]$, then the Z-transform of $x[n-k]$ is:

   $z^{-k} X(z)$

Z-Transform of Standard Sequences

1. Unit Step Function: The Z-transform of $u[n] = 1$ for $n \geq 0$ is:

   $X(z) = \frac{1}{1 - z^{-1}}, \quad |z| > 1$

2. Exponential Sequence: The Z-transform of $x[n] = a^n u[n]$ is:

   $X(z) = \frac{1}{1 - az^{-1}}, \quad |z| > |a|$

Example:

For $x[n] = 2^n u[n]$, the Z-transform is:

$X(z) = \frac{1}{1 - 2z^{-1}}, \quad |z| > 2$

Inverse Z-Transform

The Inverse Z-Transform is used to recover the original sequence from its Z-transform:

$x[n] = \frac{1}{2\pi i} \oint X(z) z^{n-1} dz$

Alternatively, partial fraction decomposition or power series expansion methods can be used.

Solving Difference Equations Using Z-Transform

The Z-transform can be applied to solve linear difference equations. Consider the equation:

$x[n+2] - 3x[n+1] + 2x[n] = 0$

Taking the Z-transform:

$z^2 X(z) - 3zX(z) + 2X(z) = 0$

Solving for $X(z)$, we can use the inverse Z-transform to find the solution in the time domain.