Fourier Analysis PDF

Chapter-3-Introduction to Fourier Analysis Introduction Fourier series, Fourier transform and their applications was given by Joseph Fourier who was French mathematician and physicist. Fourier series expansion is used for periodic signals to expand them in terms of their harmonics which are...

Chapter-3-Introduction to Fourier Analysis Introduction Fourier series, Fourier transform and their applications was given by Joseph Fourier who was French mathematician and physicist. Fourier series expansion is used for periodic signals to expand them in terms of their harmonics which are sinusoidal and orthogonal to one another. Fourier Series is like breaking down a repeating (periodic) signal into a sum of simple sine and cosine waves. Think of it as a recipe that tells you which frequencies and amplitudes are needed to reconstruct the periodic signal. Key Idea: It works only for periodic signals. The signal is represented as a sum of discrete frequency components (harmonics). Frequencies are integer multiples of the fundamental frequency 1/T​, where T is the period. Periodic signals: There are two types of periodic signals Continuous time periodic signals | Continuous time Fourier series Discrete time periodic signals | Discrete time Fourier series Examples: AC Voltage and Current: The electrical signals in power systems (e.g., 50 Hz or 60 Hz sinusoidal waveforms). Heartbeats: Under normal conditions, the electrical signals from a healthy heart (like ECG) are approximately periodic. Rotating Machines: Signals from rotating systems like engines or turbines often generate periodic waveforms. We use Fourier series expansion for the analysis purpose. We know that real life signals are non periodic in nature and for analysis of non periodic signals we need some tool and the tool was given by Joseph Fourier and it is known as Fourier transform. Non periodic signals: There are two types of non-periodic signals Continuous time non periodic signals | Continuous time Fourier transform Discrete time non periodic signals | Discrete time Fourier transform 1  Examples: Speech Signals: Words spoken by a person are non-repetitive, making speech signals non-periodic. Music and Audio: Unless it’s a pure tone or looped, most music is non-periodic. Environmental Data: Signals like temperature changes, seismic waves, and stock market data are typically non-periodic. Digital Data: Data transmitted in networks (like Ethernet or Wi-Fi) are non-periodic. Laplace transform is used for designing purposes [we obtain the transfer function using the Laplace transform and by using the different methods available we can easily check the stability of the system and by using the results obtained we can design our system] while Fourier series and Fourier transform are used for analysis purposes. Z transform is the discrete time version of Laplace Transform. Periodic Signals Periodic signals are those signals in which there is repetition of a particular structure from -∞ to ∞ or A periodic signal is a signal in which the signal repeats itself after a particular interval of time and this particular interval of time is known as time period of the signal. T is a time period, T = nT0 where T0 is the fundamental time period [the smallest time interval for which the given signal is periodic] and we are performing the time shifting [either the left shift or the right shift] but the shifting interval should be the time period so whenever we perform the shifting, we will get the same signal. The existence of Fourier series expansion depends on three conditions and the conditions are known as dirichlet conditions. Harmonics Frequency is defined as the number of cycles per second. 2  Real-life signals are typically not periodic in nature. For example, during a phone call, our voice is converted into an electrical signal by the microphone we are using. The microphone contains a transducer that transforms our voice into an electrical signal. Unlike the periodic signals described earlier, this signal depends on the words we speak. Each word generates a unique pattern, resulting in different spikes in the waveform. In this case, we cannot use the earlier formula to calculate the frequency, as it relies on identifying cycles, which is not possible here. Instead, we define frequency in a more general way: it is the rate of change in the signal. For example, saying the letter T involves rapid variations, resulting in a high rate of change (higher frequency), while saying the letter C has slower variations (lower frequency). In the periodic case, we dealt with a single frequency, but here we encounter multiple frequencies. Strictly speaking, this type of signal is referred to as a signal because it carries information, which is often unknown, non-deterministic, and random. A practical signal like this contains multiple frequencies. It can be expressed as a combination of a fundamental frequency component along with its harmonics. Let signal X(t) is a periodic signal and it is expressed as the sum of the original signal and the harmonics. The first term, sin(ωt), represents the original signal and is also known as the first harmonic because its frequency corresponds to the fundamental frequency (ω). The second term has a frequency of 2ω, which is an integral multiple of the fundamental frequency. When a frequency is an integral multiple of the fundamental frequency, we 3  refer to it as a harmonic. If we focus on the coefficients, we can observe that the effect of the 3rd harmonic is more significant than that of the 2nd harmonic. Whenever a signal contains different frequency components along with the fundamental frequency, we say that harmonics are present. This is what we aim to analyze. In the example above, we observed that the 3rd harmonic is more dominant than the 2nd harmonic. To achieve this form of signal expansion, we can use Fourier analysis. In the figure above, we see a square waveform, which contains various harmonics. The fundamental signal is a sine waveform. When we perform the Fourier series expansion of this square wave, we find that it consists entirely of sine terms, with different harmonics present in the expansion. The harmonics observed in the waveform are determined using the Fourier series. Types Trigonometric Fourier series expansion Complex exponential Fourier series expansion Polar or harmonic Fourier series expansion Dirichlet Condition: The conditions for the existence of a Fourier series were established by the German mathematician Peter Gustav Lejeune Dirichlet. According to Dirichlet's conditions, a periodic signal must be absolutely integrable over its time period. Absolutely integrable means that when you integrate the signal over a given period, the result should be finite (i.e., less than infinity). 4 As you can see in the figure above, signal x1(t) is absolutely integrable as the integral over one time period is finite(AT0/2) therefore the Fourier series expansion for signal x1(t) will exist. Trigonometric Fourier Series Fourier series expansion is used only for periodic signals, so in the formulas involved in the trigonometric Fourier series expansion we will find the formulas having the time period (T). In the case of trigonometric Fourier series expansion, for a periodic signal X(t) we can represent this periodic signal with a sum of DC or average value of signal X(t), cosine terms and sine terms. So our task is to obtain the above terms and their sum will give us the Fourier series expansion of the given periodic signal. ○ DC or average value: represents the constant (non-varying) part of the signal over a period. ○ Cosine and Sine terms: represent the oscillatory components of a periodic signal, describing how the signal varies over time. And the coefficients in this expansion can be found by ○ ω0: is angular frequency (ω0 = 2πf0), where f0 is the fundamental frequency (1 / T) ○ an: Coefficients of the cosine terms, representing the contribution of even-symmetric components at frequency nω0. ○ bn: Coefficients of the sine terms, representing the contribution of odd-symmetric components at frequency nω0. The physical significance of an and bn To understand the physical significance of an and bn lets see an example, Let's assume the below signal is periodic, 5 Let's focus on the coefficients an and bn. If you see the first two terms involving the cosine and sine you will find the frequency is equal to twice of w0, so they are the second harmonics of the signal, and if you see later two terms you will find the frequency is three times w0, therefore the terms are the third harmonics of signal. Also in the second harmonics, if you see the coefficient along with the cosine term and sine term you will find it is 3 and 5, so we can say a2 is equal to 3 and b2 is equal to 5. And if you compare them you will find a2 is less than b2. This implies the involvement of the sine term is more as compared to the cosine term when frequency is equal to twice of w0. By the same method we can find a3 and b3 to be 4 and 2 respectively. This implies the involvement of the cosine term is more as compared to the sine term when frequency is equal to three times w0. Important Points If the given periodic signal is symmetrical about the time axis (x axis) then the average value(a0) is going to be zero, because the total area will be equal to zero. If the given periodic signal is an even signal, then the coefficient bn is equal to 0. If the given periodic signal is an odd signal, then the coefficient an is equal to 0. To summarize, The Fourier series expresses any periodic function as a sum of a DC component (the average value) and oscillating sine and cosine waves of increasing frequencies. The fundamental frequency sets the base oscillation, while harmonics refine the shape, capturing details. Together, these components decompose the function into its building blocks, revealing its frequency content. Examples It is symmetrical about the time axis (a0 = 0) It is an even signal (bn = 0) It is symmetrical about the time axis (a0 = 0) It is an odd signal (an = 0) 6  It is an even signal (bn = 0) Fourier Transform The Fourier Transform is a crucial tool for signal analysis, extending the concept of the Fourier Series to non-periodic signals. The Fourier Series is typically used to decompose a periodic signal X(t), where the signal satisfies X(t+nT) = X(t) for all integers n and a period T. This decomposition expresses the signal as a sum of sines and cosines at discrete frequencies. However, most signals of interest are non-periodic, such as speech or audio signals, which do not repeat in a regular pattern. The Fourier Series is limited in this regard, as it only works for periodic signals. For non-periodic signals, the Fourier Series doesn't provide sufficient information since it focuses on the frequencies within a single period. The Fourier Transform addresses this by converting a non-periodic signal into a continuous spectrum of frequencies. This allows us to analyze non-periodic signals by examining the frequencies that are present across the entire signal, not just within a repeating period. The Fourier Transform provides a more general and powerful way to study signals that don't repeat, which are the most relevant in real-world applications. To derive the Fourier Transform for a non-periodic signal, we extend the periodicity of the signal and then apply the Fourier series approach. The process can be summarized in three simple steps: 1. Define a Periodic Extension of the Non-Periodic Signal Given a non-periodic signal X(t), we create a periodic extension Xp(t) by repeating the signal at regular intervals (shifting by T). This extension allows us to apply the Fourier series to a periodic function. 7  2. Apply Fourier Series to the Periodic Signal Decompose the periodic signal Xp(t) into its Fourier series, which involves calculating the Fourier coefficients Xn​. This is done by integrating the periodic extension of the signal over one period. 3. Take the Limit as the Period Goes to Infinity As the period T of the periodic extension grows infinitely, the frequency components become closely spaced, eventually forming a continuous spectrum. This turns the summation in the Fourier series into an integral, leading to the Fourier Transform and its inverse. The periodic extension Xp(t) becomes the original non-periodic signal X(t). The fundamental frequency f0 = 1/T​ becomes infinitesimally small, turning into a differential frequency df. The discrete frequency components nf0​become continuous frequencies f. Fourier Transform Inverse Fourier Transform This process takes a non-periodic signal, extends it periodically, applies the Fourier series, and then takes the limit to obtain the continuous Fourier Transform and its inverse. Two Worlds: Time and Frequency Domains The time domain is where we observe signals as functions of time, governed by differential equations, which are often complex and lack universal solutions. The frequency domain is the "mirror image" of the time domain, where every phenomenon in time has a counterpart in 8 frequency. Here Differential equations are converted into algebraic equations, which are easier to solve with established mathematical tools. The Fourier Transform acts as a gateway between these two domains, enabling transitions back and forth: Time to Frequency: Fourier Transform (FT) Frequency to Time: Inverse Fourier Transform (IFT) Some Examples 1. Impulse signal (Dirac delta function, δ(t)): The Fourier Transform of δ(t) is: X(f) = 1, for all f, This illustrates the inverse time-frequency relationship: faster time-domain changes require richer high-frequency content. Since δ(t) transitions infinitely fast, it spans all frequencies equally in the frequency domain. 2. Cosine signal: The Fourier Transform of cos⁡(2πf0t) reveals that it consists of two delta functions in the frequency domain, representing its spectral components at positive and negative frequencies. Mathematically: Basic properties of FT 1. Linearity (Superposition): The Fourier Transform of a linear combination of two or more signals is equal to the same linear combination of their respective Fourier Transforms. 2. Time-Shift: If a signal x(t) has a Fourier Transform X(f), then the Fourier Transform of a time-shifted version of x(t), denoted as x(t−t0), is given by: This equation states that shifting a signal in time introduces a phase shift in its Fourier Transform, while the magnitude spectrum remains unaffected. Proof ❖ Fourier Transform of the Time-Shifted Signal: 9  ❖ Let u = t − t0​. Then, t = u + t0​, and dt = du. Changing the limits of integration: ❖ Separate Exponential Terms ❖ Substituting back: ❖ Factor out the constant phase shift as it is independent of u, ❖ The integral is the Fourier Transform of x(t), which is X(f): ❖ Shifting x(t) in time by t0​does not introduce new frequencies or change their relative amplitudes. It simply alters the alignment of the frequencies in time to match the shift. 3. Scaling: The scaling property of the Fourier Transform describes how a signal x(t) is affected in the frequency domain when it is compressed or stretched in the time domain. where: a > 1: The signal is compressed in time and expanded in frequency. 0 < a < 1: The signal is stretched in time and compressed in frequency. 4. Convolution: It states that the Fourier Transform of the convolution of two signals in the time domain is the pointwise product of their Fourier Transforms in the frequency domain. Mathematically: Where x(t) and y(t) are time-domain signals, x(t)∗y(t) denotes their convolution, and X(f), Y(f) are their respective Fourier Transforms. 10  In linear systems, the output signal y(t) can be found by convolving the input signal x(t)) with the system’s impulse response h(t). Using the Fourier Transform, this process is simplified to multiplying the input signal's spectrum X(f) by the system's frequency response H(f): 5. Modulation: Let X(t) be a time-domain signal, and let X(f) be its Fourier transform. If the signal X(t) is multiplied by a cosine function cos⁡(2πf0t), the resulting signal in the frequency domain is given by: This means that multiplying the time-domain signal by a cosine function causes a shift in the frequency spectrum of the original signal. In radio communication, the voice or audio signal is typically in the range of 0 to 10 kHz (baseband signal). By modulating this signal with a cosine carrier at a higher frequency (e.g., 89.5 MHz for FM radio), the baseband signal is translated to the desired radio frequency band. This allows multiple radio stations to transmit simultaneously, each occupying a different frequency range. Examples Text Book ○ 5.5 b ○ 5.6 b ○ 5.7 b ○ 5.8 b ○ 5.19 11