Fourier Transforms

Fourier Transforms:- Time domain and frequency domain representation of the signal, Fourier Transform and its properties, conditions for existence, Transform of Gate, unit step, constant, impulse, sine and cosine wave. Shifting property of delta function, convolution, time and frequency convolution theorems.

Time domain and frequency domain representation of the signal 

An electrical signal either, a voltage signal or a current signal can be represented in two forms: Two types of representations follow: 

• Time Domain representation-: In time domain representation a signal is a time-varying quantity as shown in Figure 1.

Figure 1:- An arbitrary time domain signal

• Frequency Domain Representation: In the frequency domain, a signal is represented by its frequency spectrum as shown in Figure 2.

Figure 2:- Frequency domain representation of time domain signal

Fourier Transform and its properties

• Fourier Transform pair

Fourier transform may be expressed as

In the above equation X(w) is called the Fourier transform of x(t). In other words X(w) is the frequency domain representation of time domain function x(t). This means that we are converting a time domain signal into its frequency domain representation with the help of the Fourier transform. Conversely if we want to convert the frequency domain signal into a corresponding time domain signal, we will have to take inverse Fourier transform of the frequency domain signal. Mathematically, Inverse Fourier transform.

Example

Properties of Continuous Time Fourier Transform (CTFT)

• I. Time Scaling Function

The time scaling property states that the time compression of a signal results in its spectrum expansion

and time expansion of the signal results in its spectral compression. Mathematically,

The function x(at) represents the function x(t) compressed in the time domain by a factor a. Similarly, a function X(w/a) represents the function X(w) expanded in the frequency domain by the same factor a.

• II. Linearity Property

The linearity property states that the Fourier transform is linear. This means that

• III. Duality or Symmetry Property

• IV. Time-shifting property

• V. Frequency Shifting Property

The frequency shifting property states that the multiplication of function x(t) by is equivalent to shifting its Fourier transform X(w) in the positive direction by an amount wo. This means the spectrum X(w) is translated by an amount c. Hence this property is often called frequency frequency-translated theorem.

Mathematically.

• VI. Time Differentiation Property

The time differentiation property states that the differentiation of a function x(t) in the time domain is

equivalent to the multiplication of its Fourier transform by a factor jw. Mathematically

• 3. Transform of Gate

A gate function is a rectangular pulse. Figure 3 shows the gate function. The function or rectangular pulse shown in figure 3 is written as rect (t/𝜏).

Figure 3 :- A Gate Function

From the above figure, it is clear that rect (t/𝜏) represents a gate pulse of height or amplitude unity and width 𝜏.

Sampling Function Or Interpolation Function Or Sinc function

The function sin x/x is the "sine over argument" and is denoted by sinc(x). This function plays an important role in signal processing. It is also known as the filtering or interpolating function. Mathematically,

Sinc(x)= sin x/x

Or

Sa(x)= sin x/x

Figure 4:- Sample function

From the figure, the following points may be observed about the sampling function :

Example 2: Find the Fourier transform of the gate function shown in the figure 5.

Figure 5:- Gate function

The figure 6 shows the plot of X(w) 2π/𝜏.

Figure 6

• 4. Impulse Functions

Unit Impulse functions:

A unit impulse function was invented by P.A.M. Diarc and so it is also called as Delta function. It is denoted by δ(t).

Mathematically,

The figure shows the graphical representation of a unit impulse function. The following points may be

observed about a unit-impulse function:

Figure 7:- The Unit Impulse function

1. The width of the pulse is zero. This means that the pulse exists only at t=0.
2. The height of the pulse goes to infinity
3. The area under the pulse-curve is always is always unity.

Shifting Property of the Impulse function:

Q.1 Find the Fourier transform of an impulse function x(t) = δ(t) Also draw the spectrum

Sol. Expression of the Fourier transform is given by

Hence the Fourier transform of an impulse function is unity.

Figure 8:- Time and frequency domain of impulse function

Figure 8 shows a unit impulse function and its Fourier transforms or spectrum. From the figure, it is clear that a unit impulse contains the entire frequency components having identical magnitude. This means that the bandwidth of the unit impulse function is infinite. Also, since the spectrum is real, only the magnitude spectrum is required. The phase spectrum θ(w)=0, which means that all the frequency components are in the same phase.

• Q.(2) Find the inverse Fourier transform of δ(w)

Solution. Inverse Fourier transform is expressed as

Figure 9:- representation of inverse Fourier transform

This shows that the spectrum of a constant signal x(t)=1 is an impulse function 2πδ(w). This can also be interpreted as that x(t) =1 is a d.c. signal which has a single frequency. W=0(dc).

• Q.(3) Find the inverse Fourier transform of δ(W-Wo)

Solution. Inverse Fourier transform is expressed as

• 5. Fourier Transform of Cosine wave

Q.4 Find the Fourier transform of overlasting sinusoid cos wo𝜏􀬴􀝐.

solution: We know that Euler’s identity is given by

• 6. Fourier Transform of Periodic Function

Fourier transform of periodic function could also be found out. This means that the Fourier transform may be used as a universal mathematical tool to analyze both periodic and non-periodic waveforms over the entire interval. Let us find the Fourier transform of periodic function x(t). x(t) may be expressed in terms of complex Fourier series as

• Convolution

Convolution is a mathematical operation used to express the relation between input and output of an LTI system. It relates the input, output and impulse response of an LTI system as

y(t)=x(t)∗h(t)

• Where y (t) = output of LTI
• x (t) = input of LTI
• h (t) = impulse response of LTI

There are two types of convolutions:

1. Continuous convolution
2. Discrete convolution

• 7. Continuous Convolution

A convolution is a mathematical operation that represents a signal passing through an LTI (Linear and Time-Invariant) system or filter.

• Discrete Convolution

Delta function, symbolized by the Greek letter delta, δ[n] is a normalized impulse, that is, sample number zero has a value of one, while all other samples have a value of zero. For this reason, the delta function is called the unit impulse. The impulse response is the signal that exits a system when a delta function (unit impulse) is the input. If two systems are different in any way, they will have different impulse responses. The input and output signals are often called x[n] and y[n], the impulse response is usually given the symbol, h[n]. Any impulse can be represented as a shifted and scaled delta function.

Properties of Convolution

Note:

• The convolution of two causal sequences is causal.
• Convolution of two anti-causal sequences is anti-causal.
• The convolution of two unequal-length rectangles results in a trapezium.
• The convolution of two equal-length rectangles results in a triangle.
• A function convoluted itself is equal to the integration of that function.

Limits of Convoluted Signal

If two signals are convoluted then the resulting convoluted signal has the following range:

The sum of lower limits < t < sum of upper limits

Here, we have two rectangles of unequal length to convolute, which results in a trapezium. The range of convoluted signals is:

The sum of lower limits < t < sum of upper limits

−1+−2<t<2+2−1+−2<t<2+2

−3<t<4−3<t<4

Hence the result is trapezium with period 7.

• Area of Convoluted Signal

• DC Component

DC component of any signal is given by

DC component=area of the signal/period of the signal

• NOTE:

1. The temporal output is the temporal input CONVOLVED with the Impulse Response Function.
2. The frequency domain output is the frequency domain input MULTIPLIED by the Transfer Function.
3. The frequency domain signal is the Fourier Transform of the temporal signal

Mathematically, it must be that the FT of a convolution is a product.

• Convolution Theorems:

Convolution of signals may be done either in the time domain or frequency domain. So there are the following two theorems of convolution associated with Fourier transforms:

1. Time convolution theorem
2. Frequency convolution theorem

• Time convolution theorem: 

The time convolution theorem states that convolution in the time domain is equivalent to the multiplication of their spectra in the frequency domain. 

Mathematically, if

This is the time convolution theorem

• Frequency Convolution Theorem:

The frequency convolution theorem states that the multiplication of two functions in the time domain is

equivalent to the convolution of their spectra in the frequency domain.

Mathematically, if