Fourier Series and Fourier Transform

Contents

**Describe Fourier Series and its properties** 1

**Describe Need and Application of Fourier Series** 2

**Determine Fourier Series coefficients of a signal** 5

**Describe Wave Symmetry and Cosine representation** 6

**Explain Power representation of Fourier Series** 6

**Describe Input and Output relationship of LTI system using Fourier Series** 7

**State Dirichlet’s conditions for Convergence of Fourier Series** 9

**Describe Magnitude and Phase representation of Fourier Transform** 11

**Describe properties and existence of Fourier Transform** 12

**Describe Properties and Significance of Continuous- Time Fourier Transform(CTFT)** 12

**Describe Continuous-Time Fourier Transform of Common Signals** 13

**Explain Inverse Continuous-Time Fourier Transform** 14

**Define Hilbert Transform and its properties** 15

**Describe Frequency Spectrum of Periodic Signals** 16

**Define the relation between Frequency Response and Impulse Response** 17

**Describe Correlation of LTI Systems** 18

**Explain Correlation between Energy and Power Signals** 18

**Describe Fourier Series and its properties**

Fourier series is a mathematical representation of a periodic function in terms of a sum of sines and cosines or complex exponentials. It was developed by Joseph Fourier in the early 19th century and is widely used in various fields of science and engineering, including signal processing, control theory, and communication systems.

The Fourier series representation of a periodic function f(x) with period 2π is given by:

f(x) = a0/2 + Σ(n=1 to infinity) [an*cos(nx) + bn*sin(nx)]

where a0/2 is the DC component, an and bn are the Fourier coefficients given by the integrals of f(x) multiplied by cosine and sine functions, respectively, and n is the frequency of the sine or cosine component.

**Describe Need and Application of Fourier Series**

Fourier series is a mathematical tool used to represent periodic functions in terms of a sum of sine and cosine functions. The need for Fourier series arises in many applications where periodic signals need to be analysed or synthesised. Some of the major applications of Fourier series include:

- Signal Analysis: Fourier series is widely used in signal analysis to decompose a periodic signal into its constituent frequencies. This allows for the identification of the dominant frequency components of a signal, which can be used for filtering, modulation, and other signal processing applications.
- Audio and Music Processing: Fourier series is used in audio and music processing to analyse and synthesise sound signals. For example, Fourier series can be used to represent musical notes as a sum of sine waves with different frequencies and amplitudes, which allows for the synthesis of music and audio signals.
- Image Processing: Fourier series is used in image processing to analyse and synthesise images. For example, the Fourier series can be used to represent an image as a sum of sinusoidal functions with different frequencies and amplitudes, which can be used for image compression, filtering, and other image processing applications.
- Control Systems: Fourier series is used in control systems to analyse and design feedback control systems for various applications, such as robotics, aerospace, and automotive systems.
- Communication Systems: Fourier series is used in communication systems to analyse and design modulation schemes for various communication applications, such as radio, television, and cellular systems.

**Calculate the Fourier Series representation of Continuous-Time Periodic Signal using Exponential method**

The exponential method of finding the Fourier series representation of a continuous-time periodic signal involves expressing the signal as a sum of complex exponential functions with different frequencies. The steps to calculate the Fourier series representation of a continuous-time periodic signal using the exponential method are as follows:

- Determine the period T of the signal.
- Write the signal as a function of time t:

x(t) = x(t+T)

- Express the signal x(t) as a sum of complex exponential functions with different frequencies:

x(t) = Σ(n=-∞ to ∞) cn*e^{(jnω}_{n}^{ t)}

where cn are the Fourier coefficients and ω_{n} = 2πn/T is the angular frequency.

- To determine the Fourier coefficients, multiply both sides of the equation by e
^{(-jmω}_{m}^{ t)}and integrate over one period T:

∫(0 to T) x(t) e^{(-jmω}_{m}^{ t)} dt = Σ(n=-∞ to ∞) cn ∫(0 to T) e^{(j(ω}_{n}^{ – ω}_{m}^{)t)} dt

**Calculate the Fourier Series representation of Continuous-Time Periodic Signal using Trigonometric method**

The trigonometric method of finding the Fourier series representation of a continuous-time periodic signal involves expressing the signal as a sum of sine and cosine functions with different frequencies. The steps to calculate the Fourier series representation of a continuous-time periodic signal using the trigonometric method are as follows:

- Determine the period T of the signal.
- Write the signal as a function of time t:

x(t) = x(t+T)

- Express the signal x(t) as a sum of sine and cosine functions with different frequencies:

x(t) = a0/2 + Σ(n=1 to ∞) [an*cos(nω _{0} t) + bensin*(nω

*t)]*

_{0}where ω* _{0}* = 2π/T is the fundamental frequency, a0/2 is the average value of the signal over one period, and an and bn are the Fourier coefficients.

**Determine Fourier Series coefficients of a signal**

To determine the Fourier series coefficients of a signal, we need to use the Fourier series formula, which expresses the signal as a sum of complex exponential or trigonometric functions with different frequencies. The Fourier series formula is:

x(t) = Σ(n=-∞ to ∞) cn * e^{(j}^{n}^{ω}_{0}^{*t)}

where cn is the nth Fourier coefficient, ω_{0} is the fundamental frequency of the signal, and j is the imaginary unit.

To find the Fourier series coefficients, we need to calculate the values of cn. There are different methods for calculating the Fourier series coefficients, such as the exponential method and the trigonometric method. The choice of method depends on the form of the signal and the convenience of calculation.

Here is a general procedure for finding the Fourier series coefficients of a signal:

- Determine the period T of the signal.
- Express the signal as a periodic function of time t:

x(t) = x(t + T)

- Use the Fourier series formula to find the coefficients cn:

cn = (1/T) * ∫(0 to T) x(t) * e^{(-j}^{n}^{ω}_{0}^{*t)} dt

where j is the imaginary unit, and the integral is taken over one period T of the signal.

**Describe Wave Symmetry and Cosine representation**

Wave Symmetry:

Wave symmetry refers to the property of a waveform that describes the shape of the wave with respect to an axis or centre point. A waveform can be symmetrical, meaning that it is a mirror-image on either side of its centre point, or asymmetrical, meaning that it is not a mirror-image. There are three types of wave symmetry:

- Even Symmetry: A waveform is said to have even symmetry if it is symmetrical about the y-axis, which is also called the vertical axis. In other words, if you draw a line down the centre of the waveform, the left side will be a mirror image of the right side.
- Odd Symmetry: A waveform is said to have odd symmetry if it is symmetrical about the origin, which is where the x-axis and y-axis intersect. In other words, if you draw a line down the centre of the waveform, the left side will be a mirror image of the right side, but upside down.
- No Symmetry: A waveform is said to have no symmetry if it is neither even nor odd, which means that there is no mirror-image relationship between any part of the waveform.

Cosine Representation:

The cosine function is a mathematical function that describes a repetitive oscillation or wave. It is commonly used to represent periodic phenomena such as sound waves, light waves, and electrical signals. The cosine function can be represented in various ways, including as a series of sine and cosine terms, as a complex exponential, or as a Fourier series. In the case of a periodic waveform, the cosine representation is a way of expressing the waveform as a sum of cosine functions with different frequencies and amplitudes. The frequency domain representation of a waveform is useful for analysing the waveform’s spectral content, which is the distribution of energy across different frequency bands. The cosine representation is also used in digital signal processing to convert time-domain signals to frequency-domain signals and vice versa, using the fast Fourier transform (FFT) algorithm.

**Explain Power representation of Fourier Series**

The power representation of Fourier series is a way of representing a periodic waveform in terms of the power of its Fourier coefficients. The Fourier coefficients are the amplitudes of the sine and cosine waves that make up the periodic waveform. The power of each Fourier coefficient is equal to the square of its amplitude. Therefore, the power representation of a periodic waveform is a series of powers of the Fourier coefficients, which can be calculated using the following formula:

P = (a0^{2})/2 + ∑[(an^{2} + bn^{2})/2]

where P is the total power of the waveform, a0 is the DC component of the waveform, and an and bn are the Fourier coefficients of the sine and cosine waves, respectively.The power representation of Fourier series is useful for analysing the power distribution of a periodic waveform across different frequency components. The power spectrum of a waveform is a plot of its power as a function of frequency. The power spectrum can be obtained by calculating the power of each Fourier coefficient and plotting it as a function of the corresponding frequency. The power spectrum provides information about the spectral content of the waveform, including the dominant frequencies and their power levels.

The power representation of Fourier series is also used in engineering applications, such as in the design of filters and equalisers. For example, a high-pass filter attenuates low-frequency components of a waveform while allowing high-frequency components to pass through. The power representation of the waveform can be used to design a filter that selectively attenuates certain frequency components while preserving others.

**Describe Input and Output relationship of LTI system using Fourier Series**

Linear Time-Invariant (LTI) systems are a class of systems in which the output is a linear function of the input and the system parameters do not change with time. The input-output relationship of an LTI system can be described using Fourier series, which is a representation of a periodic waveform in terms of a sum of sine and cosine functions.

In the frequency domain, the input and output signals of an LTI system can be represented as complex exponential signals. The input signal can be expressed as:

x(t) = ∑[Xk * e^{(j2πkft)}]

where Xk is the complex Fourier coefficient of the input signal at frequency fk, j is the imaginary unit, f is the fundamental frequency of the waveform, and t is time.

Similarly, the output signal can be expressed as:

y(t) = ∑[Yk * e^{(j2πkft)}]

where Yk is the complex Fourier coefficient of the output signal at frequency fk.

**Explain Gibbs Phenomenon**

Gibbs phenomenon is a phenomenon that occurs when a periodic waveform is approximated by a truncated Fourier series, which is a sum of a finite number of sine and cosine waves. The phenomenon is characterised by the appearance of overshoots or ringing near the edges of the waveform, particularly at the points of discontinuity.

The Gibbs phenomenon is a result of the Gibbs phenomenon in mathematics, which states that the Fourier series of a function that has a jump discontinuity will converge to the midpoint of the jump with an overshoot of about 9% of the size of the jump. In other words, even if the number of terms in the Fourier series is increased, the overshoot will not disappear and will always be present.

In the context of signal processing, the Gibbs phenomenon can result in distortion of the waveform and introduce unwanted frequency components in the frequency spectrum. This can be particularly problematic in applications such as audio and image processing, where the quality of the waveform is important.

One approach to reducing the Gibbs phenomenon is to use a different type of Fourier series, such as the complex exponential form of Fourier series, which has better convergence properties. Another approach is to use a smoothing function, such as a window function, to gradually taper the waveform near the points of discontinuity. This can help to reduce the overshoots and ringing in the waveform.

**State Dirichlet’s conditions for Convergence of Fourier Series**

Dirichlet’s conditions are a set of conditions that must be satisfied for a periodic waveform to have a convergent Fourier series representation. The conditions are as follows:

- Periodic: The waveform must be periodic, meaning that it repeats itself after a fixed interval of time.
- Piecewise continuous: The waveform must be piecewise continuous, meaning that it can have a finite number of finite discontinuities within one period. This means that the waveform can have a finite number of points where it is not continuous, but these points must be isolated and cannot be infinitely close to each other.
- Absolute integrability: The waveform must be absolutely integrable, meaning that the integral of the absolute value of the waveform over one period must be finite.

If a periodic waveform satisfies these conditions, then its Fourier series representation converges to the original waveform in the mean, or in other words, the Fourier series approaches the waveform as the number of terms in the series increases.

Dirichlet’s conditions ensure that the Fourier series representation of a periodic waveform is well-defined and has meaningful physical interpretations. They also ensure that the Fourier series representation accurately captures the essential features of the waveform, including its frequency content and amplitude modulation.

**Describe Fourier Transform**

The Fourier Transform is a mathematical tool used to represent a time-domain signal as a sum of sinusoidal functions in the frequency domain. It is a way of decomposing a complex waveform into its constituent frequency components, and is used extensively in signal processing, communications, and other areas of science and engineering.

The Fourier Transform takes a continuous-time or discrete-time signal in the time domain and converts it into a continuous or discrete frequency spectrum. The Fourier Transform of a time-domain signal x(t) is defined as:

X(f) = ∫ x(t) * e^{(-j2πft)} dt

where X(f) is the frequency-domain representation of the signal, f is the frequency variable, t is the time variable, and j is the imaginary unit.

The Fourier Transform of a discrete-time signal x[n] is defined as:

X(f) = ∑ x[n] * e^{(-j2πfn/N)}

where N is the length of the signal, and f and n are frequency and time index variables, respectively.

The Fourier Transform provides a way to analyse and manipulate signals in the frequency domain. The magnitude and phase of each frequency component of the signal can be extracted from the Fourier Transform, allowing for selective filtering, amplification, or attenuation of specific frequency bands. Additionally, the Fourier Transform can be used to solve differential equations, analyse complex systems, and perform other mathematical operations.

**Describe Magnitude and Phase representation of Fourier Transform**

The Fourier Transform of a signal is a complex-valued function that has both a magnitude and a phase component. The magnitude and phase of the Fourier Transform provide information about the frequency content and phase relationships of the original signal.

The magnitude of the Fourier Transform, denoted as |X(f)|, represents the amount of energy or power in each frequency component of the signal. It is a non-negative real-valued function, and its value at a particular frequency f represents the amplitude of the sinusoidal component at that frequency. The magnitude spectrum of a signal is often plotted on a logarithmic scale, known as the magnitude spectrum or frequency spectrum.

The phase of the Fourier Transform, denoted as Φ(f), represents the phase shift or time delay of each frequency component of the signal. It is a real-valued function, and its value at a particular frequency f represents the phase shift or time delay of the sinusoidal component at that frequency. The phase spectrum of a signal is often plotted as a function of frequency.

The complex Fourier Transform X(f) can be expressed in terms of its magnitude |X(f)| and phase Φ(f) as follows:

X(f) = |X(f)| * e^{(jΦ(f))}

where j is the imaginary unit. The magnitude and phase components of the Fourier Transform can be extracted using the following formulas:

|X(f)| = sqrt(Re(X(f))^{2} + Im(X(f))^{2})

Φ(f) = a tan2(Im(X(f)), Re(X(f)))

**Describe properties and existence of Fourier Transform**

The Fourier Transform is a powerful mathematical tool that is widely used in signal processing, communications, and other areas of science and engineering. It has many important properties that make it useful for analysing and manipulating signals in the frequency domain.

- Linearity: The Fourier Transform is a linear operator, meaning that it satisfies the properties of additivity and homogeneity. This means that the Fourier Transform of a sum of signals is equal to the sum of their individual Fourier Transforms, and that the Fourier Transform of a scaled signal is equal to the scaled Fourier Transform of the original signal.
- Time Shifting: The Fourier Transform of a time-shifted signal is equal to the Fourier Transform of the original signal multiplied by a phase factor that depends on the amount of time shift. This property is often used to analyse the phase relationships between different frequency components of a signal.
- Frequency Shifting: The Fourier Transform of a frequency-shifted signal is equal to the Fourier Transform of the original signal multiplied by a complex exponential function that depends on the amount of frequency shift. This property is often used to analyse the effect of changing the centre frequency of a signal.
- Convolution: The Fourier Transform of the convolution of two signals is equal to the pointwise product of their Fourier Transforms. This property is often used to analyse the frequency response of linear time-invariant systems, which can be represented as convolution operations.

**Describe Properties and Significance of Continuous- Time Fourier Transform(CTFT)**

The Continuous-Time Fourier Transform (CTFT) is a mathematical tool used to represent a continuous-time signal in the frequency domain. It is a generalisation of the Discrete-Time Fourier Transform (DTFT) for continuous-time signals. The CTFT has many properties that make it useful for analysing continuous-time signals.

- Linearity: The CTFT is a linear operator, meaning that it satisfies the properties of additivity and homogeneity. This means that the CTFT of a sum of signals is equal to the sum of their individual CTFTs, and that the CTFT of a scaled signal is equal to the scaled CTFT of the original signal.
- Time Shifting: The CTFT of a time-shifted signal is equal to the CTFT of the original signal multiplied by a phase factor that depends on the amount of time shift. This property is often used to analyse the phase relationships between different frequency components of a signal.
- Frequency Shifting: The CTFT of a frequency-shifted signal is equal to the CTFT of the original signal multiplied by a complex exponential function that depends on the amount of frequency shift. This property is often used to analyse the effect of changing the centre frequency of a signal.
- Duality: The CTFT has a duality property, which means that if we take the CTFT of a signal and then invert it using the inverse CTFT, we get the original signal back. This property is important because it allows us to analyse signals in both the time and frequency domains.
- Time and Frequency Resolution: The CTFT allows us to analyse the frequency content of a signal with arbitrary time resolution. However, there is a fundamental tradeoff between time and frequency resolution, known as the uncertainty principle. This means that we cannot simultaneously have perfect time and frequency resolution.

The significance of the CTFT is that it allows us to analyse continuous-time signals in the frequency domain, which can be very useful for understanding the behaviour of signals in communication systems, audio processing, and other applications. The CTFT is also the basis for other frequency analysis tools, such as the Laplace transform and the Z-transform.

**Describe Continuous-Time Fourier Transform of Common Signals**

The Continuous-Time Fourier Transform (CTFT) is a mathematical tool used to represent a continuous-time signal in the frequency domain. Here are the CTFTs of some common signals:

- Impulse function: The CTFT of an impulse function, also known as a Dirac delta function, is a constant function with value 1 for all frequencies:

F(jω) = 1

- Sinusoidal function: The CTFT of a sinusoidal function with frequency ω0 is two delta functions located at ±ω0:

F(jω) = 2π [δ(ω – ω0) + δ(ω + ω0)]

- Cosine function: The CTFT of a cosine function with frequency ω0 is also two delta functions located at ±ω0, but with opposite phase:

F(jω) = 2π [δ(ω – ω0) + δ(ω + ω0)] cos(θ)

where θ is the phase angle of the cosine function.

**Explain Inverse Continuous-Time Fourier Transform**

The inverse continuous-time Fourier transform (ICTFT) is the mathematical operation that allows us to recover a continuous-time signal from its frequency domain representation given by the Continuous-Time Fourier Transform (CTFT). The inverse transform is defined as follows:

x(t) = (1/2π) ∫[-∞,∞] X(f) e^{(j2πft)} df

where X(f) is the CTFT of the signal x(t). This means that if we know the frequency domain representation of a signal, we can use the ICT FT to obtain the original signal in the time domain.

The ICTFT is also known as the Fourier synthesis formula because it allows us to synthesise a signal by adding up the contributions of its frequency components. It can be shown that the CTFT and DTFT are inverse operations of each other, which means that applying the CTFT to a signal and then applying the ICF to the resulting frequency domain representation should yield the original signal.

The ICTFT has several properties that are similar to those of the CTFT, including linearity, time shifting, and frequency shifting. It also satisfies Parseval’s theorem, which states that the energy of a signal in the time domain is equal to the energy of the signal in the frequency domain.

**Define Hilbert Transform and its properties**

The Hilbert transform is a mathematical operator that is used in signal processing, physics, and engineering. It is named after the German mathematician David Hilbert, who introduced the concept in 1905.

The Hilbert transform H(x(t)) of a real-valued function x(t) is defined as follows:

H(x(t)) = 1/πpv ∫_{-\infty}^{\infty} x(τ)/(t-τ) dτ

where “pv” denotes the Cauchy principal value of the integral.

Linearity: The Hilbert transform is a linear operator, which means that it satisfies the following properties: H(axe*(t) + b*y(t)) = a*H(x(t)) + *H(y(t)) where “a” and “b” are constants and x(t) and y(t) are functions.

- Analyticity: The Hilbert transform converts a real-valued signal into a complex-valued analytic signal, which has the following properties:a. Its real part is equal to the original signal x(t).b. Its imaginary part is the Hilbert transform of x(t).c. Its Fourier transform has non-zero values only in the upper or lower half-plane, depending on the sign of the imaginary part.
- Shift invariance: The Hilbert transform is shift-invariant, which means that if x(t) is shifted by a time constant τ, then its Hilbert transform is also shifted by the same time constant τ.
- Conjugate symmetry: The Fourier transform of the Hilbert transform has conjugate symmetry, which means that its positive and negative frequency components are complex conjugates of each other.
- Causality: The Hilbert transform is causal, which means that its output at any time t depends only on the input signal up to time t. However, its impulse response is not causal.

**Describe Frequency Spectrum of Periodic Signals**

A periodic signal is a signal that repeats itself after a fixed time interval, known as the period. The frequency spectrum of a periodic signal describes the frequency content of the signal, which is the set of frequencies present in the signal and their amplitudes and phases.

The frequency spectrum of a periodic signal can be obtained using the Fourier series representation of the signal, which expresses the signal as a sum of sinusoidal components of different frequencies. The Fourier series representation of a periodic signal x(t) with period T can be written as:

x(t) = a0 + ∑_{n=1}^{\infty} (an*cos(nω0*t) + bensin*(nω0*t))

where ω0 = 2π/T is the fundamental frequency, an and bn are the Fourier series coefficients, and a0 is the DC component.

The Fourier series coefficients an and bn can be computed using the following equations:

an = (2/T) ∫_{-T/2}^{T/2} x(t)*cos(nω0*t) dt

bn = (2/T) ∫_{-T/2}^{T/2} x(t)*sin(nω0*t) dt

The frequency spectrum of the periodic signal x(t) is then given by the complex Fourier series coefficients c_n = an – jbn, where j is the imaginary unit. The complex Fourier series coefficients c_n represent the amplitude and phase of the sinusoidal component with frequency nω0.

**Define the relation between Frequency Response and Impulse Response**

The frequency response and impulse response are two important characteristics of a linear time-invariant (LTI) system.

The frequency response of an LTI system is a complex-valued function that describes how the system responds to sinusoidal inputs of different frequencies. It is obtained by taking the Fourier transform of the system’s impulse response, which is the system’s output when its input is an impulse function.

The impulse response of an LTI system is a function that describes the system’s output when its input is an impulse function. It is related to the frequency response by the Fourier transform:

H(f) = ∫{-∞}^{∞} h(t)*e^{-j2πft} dt

where H(f) is the frequency response of the system, h(t) is the impulse response of the system, and j is the imaginary unit.

The frequency response H(f) is a complex-valued function that represents the system’s gain and phase shift for each frequency component of the input signal. The magnitude of the frequency response, |H(f)|, represents the gain of the system at each frequency, while the phase of the frequency response, arg(H(f)), represents the phase shift of the system at each frequency.

**Describe Correlation of LTI Systems**

The correlation of an LTI (linear time-invariant) system is a measure of how much the output of the system is similar to its input, but delayed in time. It is a useful tool for analysing the behaviour of LTI systems, especially in the context of signal processing.

The correlation of an LTI system can be computed using its impulse response. The impulse response of an LTI system is the output of the system when its input is an impulse function. If we denote the impulse response of an LTI system by h(t), then the correlation of the system at time τ can be defined as:

R(τ) = x(t)h(t-τ) dt

where x(t) is the input signal to the system.

The correlation function R(τ) is a measure of how much the output of the system at time τ is similar to the input signal x(t), but delayed in time by τ. If R(τ) is large and positive, it means that the output of the system at time τ is similar to the input signal, but delayed in time by τ. If R(τ) is negative, it means that the output of the system at time τ is opposite in polarity to the input signal, but delayed in time by τ.

**Explain Correlation between Energy and Power Signals**

In signal processing, energy and power are two important characteristics that can be used to describe a signal. Energy is a measure of the total amount of energy in a signal, while power is a measure of the rate at which energy is being transmitted.

The correlation between energy and power signals can be understood through the mathematical definitions of energy and power. The energy of a signal x(t) over a finite time interval T is defined as:

E = ∫(|x(t)|^2)dt from 0 to T

where |x(t)|^2 is the magnitude squared of the signal at time t.

On the other hand, the power of a signal x(t) is defined as the average value of the signal’s instantaneous power over time:

P = lim(T->inf) {1/T ∫(|x(t)|^2)dt from 0 to T}

We can see that power is directly proportional to energy, as power is calculated by dividing the energy of the signal by the duration of the signal. Therefore, a signal with a higher energy will have a higher power, assuming the duration of the signal remains constant. Similarly, a signal with a lower energy will have a lower power.

However, it’s important to note that not all signals have both energy and power. A signal is said to have energy if the total energy of the signal is finite, which means that the signal decays to zero as time approaches infinity. A signal is said to have power if the average power over an infinite time interval is finite, which means that the signal does not decay to zero as time approaches infinity. Therefore, a signal can have energy but not power (e.g. a finite duration pulse signal), power but not energy (e.g. a sinusoidal signal), or both energy and power (e.g. a finite energy pulse train).