Signals and Systems

Resources:

• https://www.youtube.com/watch?v=x5qRAihZRks&list=PL9RcWoqXmzaIG-RWneeqDJ-FCt66S15pl&index=2 (Hindi)

• https://www.youtube.com/playlist?list=PLXOYj6DUOGrrAlYxrAu5U2tteJTrSe5Gt (Problem Solving)

• Euler Equation :
  https://www.physicsforums.com/threads/solve-e-frac-i-pi-4.336356/
  https://proofwiki.org/wiki/Euler's_Formula/Examples/e^i_pi_by_4

Skipped:

• https://www.youtube.com/watch?v=Rew03iHJGhk&list=PL9RcWoqXmzaIG-RWneeqDJ-FCt66S15pl&index=9&pp=iAQB (Complex Exponentiation)

Signals

Physical quantity that contains information. Signals are expressed mathematically as function of independent variable, which is usually time.

Example:

Basic Types of Signals:

Based on Continuous and Discrete:

• Continuous in Time Signal & Continuous in Value Signal
  • A continuous-time signal has values for all points in time in some interval.
  • A continuous-value signal is all possible value within an interval will be available in a signal.

  

• Continuous in time but discrete in value signal
  • A continuous-time signal has values for all points in time in some interval.
  • All values within a range is not available in the signal.

  

• Continuous in value but discrete in time signal
  • Haven’t values for all points in time within an interval
  • All values within an range are available in a signal

    

• Discrete in time and discrete in value signal

  

Analog Signal: Continuity in any of the domain (time or value)

Digital Signal: Discrete in both time and value

Based on Causal, Anti-Causal, Non-Causal

• Causal Signals:
  • 0 for all negative value/time
  • $x(t) = \begin{cases} x(t) \gt 0 & t \ge 0 \\ 0& t < 0 \end{cases}$

• Non-Causal Signals
  • A signal that have positive amplitude for both positive and negative instance of time

• Anti-Causal Signal
  • 0 for all positive value/time
  • $x(t) = \begin{cases} x(t)>0 & t \le 0 \\ 0& t \gt 0 \end{cases}$

Operations on Signals

• Time Shifting Operation

  $f(t)$ is be given. $f(t \pm t_0) = ?$

  • $t_0$ is a constant
  • $+$ → advance : Shift the signal towards left by $t_0$
  • $-$ → delay : Shift the signal towards right by $t_0$
  • Amplitude doesn’t change for shifting

  

• Time Scaling Operation

  $x(t)$ is given. $f(\alpha t) = ?$

  • $\alpha$ : scaling factor
  • $\alpha > 1$ → Signal Compression (Increasing Speed) : Divide the existing limit by $\alpha$
  • $\alpha < 1$ → Signal Expansion (Decreasing Speed) : Divide the existing limit by
  • Amplitude doesn’t change for this operation

  

• Time Reversal or folding Operation

  $x(t)$ is given. $x(-t) = ?$

  • The sign of the limit will be changed

    

Example on Operation:

Elementary Signals

• Unit Step Signal
  • Also known as Heaviside Step Function
  • Unit step function is denoted by $u(t)$

    $u(t) = \begin{cases} 1 & t\ge 0 \\ 0 & t < 0 \end{cases}$

  

  • Amplitude = coefficient of $u(t)$
  • Non-Causal Signal$*$Unit Step Function $=$ Causal Signal
  • Operations on Unit Step Signal:

    

    

  • Example:

    

  

  

  • Expressing by unit step function

  

• Unit Ramp Signal
  • $r(t) = \begin{cases} t & t\ge 0 \\ 0 & t \lt 0 \end{cases}$
  • Slope = Coefficient of $r(t)$

  

  • $r(t) = t \cdot u(t)$
  • $\frac{d}{dt} r(t) = u(t)$
  • $\frac{d}{dt} [A \cdot r(t)] = A \cdot u(t)$
  • Expressing into Ramp Signal

    

• Impulse Function
  • An ideal impulse signal is a signal that is zero everywhere but at the origin (t = 0), it is infinitely high. Although, the area of the impulse is finite.
  • $\delta(t) = \begin{cases} 1 & t = 0 \\ 0 & t \ne 0 \end{cases}$
  • $A \cdot \delta(t),$ Here $A$ is the area of this impulse function.

  

  • $\int_{-\infin}^{\infin} \delta(t) = 1$ or $\int_{0^-}^{0^+} \delta(t) = 1$
  • $\int_{a}^{b} \delta(t-t_0),$ it exists $if \space a \le t_0 \le b$, otherwise $0$
  • $\delta(t) = \frac{d}{dt} u(t)$
  • Operations:

    

    

  • Properties:
    • Time Scaling Property:

      $\delta(\alpha t)$ = $\frac{1}{|\alpha|} \delta(t)$

      

    • Product Property

      $x(t) \cdot \delta(t) = x(0)\cdot\delta(t)$

      $x(t) \cdot \delta(t - t_0) = x(t_0) \cdot\delta(t-t_0)$

      

    • Shifting Property

      $(i) \int_{-\infin}^{\infin} x(t) \cdot \delta(t)\\ \\ \space \\ \\ = \int_{-\infin}^{\infin} x(0) \cdot \delta(t)\\ \\ \space \\ = x(0) \int_{-\infin}^{\infin}\cdot \delta(t) \\ \space \\ \\= x(0)$

      $(ii) \int_{-\infin}^{\infin} x(t) \cdot \delta(t-t_0)\\ \\ \space \\ \\ = \int_{-\infin}^{\infin} x(t_0) \cdot \delta(t)\\ \\ \space \\ = x(t_0) \int_{-\infin}^{\infin}\cdot \delta(t) \\ \space \\ \\= x(t_0)$

      

      

      

    • Extension Properties

      $\int_{a}^{b} x(t) \cdot\delta(g(t)) dt \\ \space \\ \delta(g(t)) = \frac{\delta(t-t_0)}{|g'(t_0)|} [if \space g \space has \space a \space real \space root \space at \space t=t_0]$

      If $g$ has more than one real root at $t_0, t_1,..., t_i$

      $s(g(t)) = \sum_{i} \frac{\delta(t-t_i)}{|g'(t_i)|}$

      Example:

      $I = \int_{-10}^{10} (t^2 + 10) \cdot \delta(t^2 - 16) dt \\ \space \\ t^2-16, t = +4, -4 \\ \space \\ I = \int_{-10}^{10} (t^2 + 10) \cdot \frac{\delta(t-4)} {|2\cdot(-4)|} dt + \int_{-10}^{10} (t^2 + 10) \cdot \frac{\delta(t+4)}{|2\cdot(4)|} dt \\ \space \\ = \frac{13}{4}$

• Exponential Function

  $x(t) = Ae^{(\alpha t)}$

  $A =$ Amplitude at $t = 0$

  

Signal Classification on Even and Odd

• Even/Symmetric Signal

  $x(-t) = x(t)$

  Example : $x(t) = \begin{cases} 2 & -a \le t \le a \\ 0 & otherwise \end{cases} , x(t) = t^2,..., t^{2n}$

• Odd/Anti-symmetric Signal

$x(-t) = -x(t)$

Example: $x(t) = t^3, t^5, .., t^{2n + 1}$

Check a function even or odd:

$x(t)$ is given. Find $x_1(t) = x(-t) = ...$

if $x(1) == x_1(1)$, it is even, otherwise odd.

Converting to Odd or Even Signal

To convert any arbitrary signal which is neither even nor odd into equivalent even or odd parts:

$x_e (t) = \frac{x(t) + x(-t)}{2}$

$x_o(t) = \frac{x(t) - x(-t)}{2}$

• $x(t)$ is complex

  $x(t) = a + ib$

  • Even

    $x(t) = \overline{x(-t)}$

  • Odd

    $x(t) = - \overline{x(-t)}$

  Conjugate symmetric Part of $x(t)$

  $x_e(t) = \frac{x(t) + \overline{x(-t)}}{2}$

  Conjugate anti-symmetric part of $x(t)$

  $x_o(t) = \frac{x(t) - \overline{x(-t)}}{2}$

• Some Operations

  $E = Even , O = Odd \\ \space \\ O \pm O = O \\ \space \\ E \pm E = E \\ \space \\ O \pm E = Neither \space Odd \space nor \space Even \\ \space \\ E \times E = E \\ \space \\ E \times O = O \\ \space \\ O \times O = E \\ \space \\ E/E = E \\ \space \\ O/O = E \\ \space \\ E/O = O \\ \space \\ O/E = O \\ \space \\ \int O = E \\ \space \\ \int E = O \\ \space \\ \frac{d}{dt} E = O \\ \space \\ \frac{d}{dt} O = E$

Periodic and Aperiodic Signal

• Periodic Signal
  • A signal is said to periodic, if it satisfies following two properties
    • It must be exist for $-\infin \le t \le \infin$
    • It must repeat itself after some constant amount of time $T$, which is called Fundamental Time Period
      • $T = \frac{2 \pi}{w_0}$
      • $w_0 =$ Fundamental frequency $rad \space s^{-1} = 2\pi f$
      • $T = \frac{1}{f}$
      • Frequency must be real number. If frequency is not real number, it’s not periodic.
  • Types
    • Sinusoidal Signals
      • Representation

        $x(t) = A sin(w_0t + \theta)$

        $A =$ Amplitude

        $w_0t =$ Phase Angle

        $\theta =$Phase shift (+ → Advance, - delay)

      • $\frac{d}{dt} (Phase) = \frac{d}{dt} (w_0t) = w_0 = freequency$
      • Shifting effect doesn’t effect on periodicity, $T$
      • $x(t \pm kT) = x(t)$
      • Comparing with $x(t) = A sin(n\pi t + \theta)$, if $t$ is not square root of $t$, then the signal must be periodic.

    

    • Combination of periodic signals

      $x(t) = Asin(w_0t) + Bcos(w_1t) + Csin(w_2t) + ....$

      • $x(t)$ will be periodic if ratio of individual time period is a rational number
        • Rational Number
          • Can be expressed by $\frac{p}{q}$. $p,q$ are co-prime.
          • The value of $\frac{p}{q}$ should terminating or repeating decimal.
            Example : 3.3333.., 2.5, 5.20202020..

        Example: $\frac{T_2}{T_1} = \frac{T_3}{T_2} = \frac{T_1}{T_3} = Rational \space Number$

      • Time Period of Resultant Signal
        $T = LCM(T_1, T_2, ....)$
      • Frequency of Resultant Signal
        $w'_0 = GCD(w_0, w_1, ....)$
      • If all $w_0, w_1, ...$ have $\pi$, then it is periodic. Or if all $w_0, w_1, ....$ haven’t $\pi$, it is periodic too. If some of them have $\pi$ and rest of them not, then it isn’t periodic.

      

      • DC/Constant Signal : independent from time. It doesn’t effect on frequency, time. So, it is not countable.

      

Energy and Power Signals

• Energy Signals

  Energy of $x(t)$ is given as

  $E_x = \lim\limits_{t \to \infin} \int_{-\frac{T}{2}}^{\frac{T}{2}} | x(t) | ^2 \space dt \\ \space \space [\text{No need to take lim if the signal is aperiodic}]$

  if $0 < E_x < \infin$ (Finite), then $x(t)$ is said to be $Energy \space Signal.$

  Example:

  $(i) \space x(t) = e^{-4t} \cdot u(t) \\ \space \\ E_x = \int_{-\infin}^{\infin} [e^{-4t}\cdot u(t)]^2 \space dt\\ \space \\ = \int_{0}^{\infin} e^{-8t} dt \space \space [\text{u(t) exists only 0 to } \infin ] \\ \space \\ =\frac{1}{8}$

  

  • When a signal $x(t)$ will be energy signal
    • If $x(t)$ is existing for infinite direction and decreasing in value.

      $\lim\limits_{t \to \infin} f(t) = 0$

    • If $x(t)$ exists for finite direction and value of $x(t)$ is finite finite at all points, $x(t)$ is energy signal.

• Power Signals

  $Power \space Signal$ of $x(t)$ is given as

  $P_x =\lim\limits_{T \to \infin} \frac{1}{T} \cdot E \\ \space \\ = \lim\limits_{T \to \infin} \frac{1}{T} \int_{-\frac{T}{2}}^{\frac{T}{2}} | x(t) | ^2 dt \\ \space \\ = \lim\limits_{T \to \infin} \frac{1}{2T} \int_{-T}^{T} | x(t) | ^2 dt \\ \text{[No need to take lim if the signal is aperiodic]}$

  If $0 < P_x < \infin$ (Finite), then $x(t)$ is said to be $\text{Power Signal}$

  • $\text{Power} = RMS^2$
  • $x(t) = A sinwt, \space RMS = \frac{A}{\sqrt{2}}, P = RMS^2 = \frac{A^2}{2}$

  

  • When a signal will be Power Signal
    • All periodic signal are power signal but converse is not true
    • If $x(t)$ is not a periodic signal and follows the conditions
      • $\lim\limits_{t \to \infin} f(t) \ne 0$
      • $\lim\limits_{t \to \infin} f(t) \ne \infin$

• A signal can’t be Energy and Power Signals together.
  • If $E_x$ is finite, then $P_x$ is $\text{Zero}$ . Vice-Versa.

• Operations
  • Time shifting has no effect on power and energy of signal.$Power \space x(t) = Power \space x(t - \frac{T}{2}) \\ Energy\space x(t) = Energy\space x(t - 4)$
  • Time Scaling doesn’t effect on Time Periodic but in Time period, Energy.
    • $\text{For }x(t) T, E_x \\ \space \\ x'(t) = x(\alpha t), T' = \frac{T}{\alpha}, E_x' = \frac{E_x}{\alpha}$
    • Power remains same.

Systems

System is a interconnection of different physical components which is used to convert one form of signal to others.

Example:

Properties of Systems

• Static and Dynamic System
  • Static
    • Memoryless
    • if present o/p depends on only present i/p

    

  • Dynamic
    • With Memory
    • if present o/p depends on past or future i/p

    

    $y(2) = x(2^2) = x(4)$, $t=2$’s output depends of $t=4$. That means present o/p depends on future i/p;

• Causal, Noncausal, Anticausal
  • Causal
    • Present o/p depends on
      • present i/p or
      • Present + Past i/p
    • Static system are always causal
    • $h(t) = 0 \space\space\space\space\space\space\space t < 0 \\ h[n] = 0 \space\space\space\space\space\space\space t < 0 \\$
  • Noncausal
    • Present o/p depends on
      • present + future or
      • present + past + future or
      • past + future
    • $h(t) \not= 0, \space \space \space t < 0 \\ h[n] \not= 0, \space \space \space t < 0 \\$

    $y[n] = x[n^2] \\ y[0] = x[0] \\ y[1] = x[1] \text{ depends on present} \\ y[2] = x[4] \text{ depends on future}$

  • Anticausal
    • Present o/p depends on
      • only future i/p

      $y(t) = x(t^2 + 1) \\ y(0) = x(1) \\ y(-1) = x(0)$

      depends on only future

    • $h(t) = 0 \space \space \space t > 0 \\ h[n] = 0 \space \space \space t > 0$

• Time variant and Invariant System
  • Time Invariant
    • If time shift in i/p results identical time shift in o/p without changing the nature of the output.
  • To check
    • Find $y(t)$ delay, $y(t-t_0)$, replace $t$ with $t-t_0$
    • Find $x(t-t_0) = y(t, t_0)$, response of the system for delayed i/p

      $y(t, t_0) = \text{write the y(t) just add/minus } t_0$

    • if $y(t-t_0) = y(t, t_0),$ it is time variant.

    

    

    

  • For a system to be time invariant
    • There must not be any scaling in $x(t) \text{ or } y(t)$
    • Coefficient must not function of $time$
    • Any extra term except $x(t) \text{ or } y(t)$ must be zero or constant

• Linear and Nonlinear System
  • Linear
    • If the system follows principle of superposition

    

  • To check linearity
    • Graph between output and input must be throughout a straight line passing through origin without having saturation or dead zone
    • If system is represented by a linear differential equation, then the equation must be linear.
    • System must be follow zero input zero output criteria.
      • $x(t) \rarr y(t) \\ \alpha x(t) \rarr \alpha y(t) \\ 0 = 0, \alpha = 0$
  • Mathematical way to prove a system is linear or not

    $\text{Given, } x(t) \rarr y(t) = tx(t)$

    $x_1(t) \rarr y_1(t) = tx_1(t)$

    $x_2(t) \rarr y_2(t) = tx_2(t)$

    $y_3(t) = y_1(t) + y_2(t) = t(x_1(t) + x_2(t)) ... (1)$

    $x_3'(t) = x_1(t) + x_2(t)$

    $y_3'(t) = tx_3(t) = t(x_1(t) + x_2(t)) ... (2)$

    $(1) =(2), \text{ it follows the additivity.}$

    $\alpha x(t) \rarr \alpha y(t) = \alpha tx(t)$

    $y'(t) = t [\alpha x(t)] = \alpha t x(t) ... (3)$

    $\alpha y(t) = \alpha t x(t) ... (4)$

    $(3) =(4), \text{it follows the homogeneity rule}$

  • Checking homogeneity and additivity at once

    $x(t) \rarr y(t) \\ \alpha x_1(t) + \beta x_2(t) \rarr \alpha y_1(t) + \beta y_2(t), \text{the system will be lineary}$

    $\text{Given, } y(t)=x[sint]$

    $y_1(t) = x_1[sint] , y_2[t] = x_2[sint]$

    $y_3(t) = \alpha y_1(t) + \beta y_2(t) = \alpha x_1(sint) + \beta x_2(sint)$

    $x_3'(t) = \alpha x_1(t) + \beta x_2(t)$

    $y_3'(t) = x_3'(sint) = \alpha x_1(sint) +\beta x_2(sint)$

• Invertible and Non-invertible
  • Invertible
    • If distinct input produces distinct outputs
    • if input can be determined by observing output
    • a inverse system can be design, overall gain = 1

• Stable and Unstable System
  • Stable
    • Follows bounded input bounded output(BIBO)
    • The output of the system must be bounded for bounded input
      • $0 \le |x(t)| < \infin$, then $0 \le |y(t)| < \infin$
    • BIBO implies that impulse response must tend to zero , as time tends to infinity.

• LTI System
  • Properties
    • The commutative property of LTI System
      • Discrete Time Signal

        $x[n] *h[n] = h[n]*x[n] \\ \text{or, }$$\sum_{k=-\infin}^{\infin} x[k]h[n-k] = \sum_{k=-\infin}^{\infin}h[k]x[n-k]$

      • Continuous Time Signal

        $x(t)*h(t) = x(t)*h(t)\\ \text{or, } \int_{-\infin}^{\infin} x(\tau)h(t-\tau) = \int_{-\infin}^{\infin} x(t - \tau)h(\tau)$

    • The distributive Property of LTI System

      Summing of the outputs of of two systems is equivalent to a system with an impulse response equal to the sum of the impulse response of the two individual system.

      

      $y_1 = y_2$

      $y_1 = x*h_1 + x*h_2$

      $y_2 = x*(h1+h2)$

      Hence, $x*h_1 + x*h_2 = x*(h_1+h_2)$

    • Associative Property

      The change of order of the cascaded system will not affect the response.

    

    $y = x*(h1*h2) = (x*h_1)*h_2$

    • LTI System with and without memory

      Memoryless → Output(t) → input(t), Current output depends on only current input

      DTS: $h[n] = 0 \text{ for } n \not= 0$

      CTS: $h[t] = 0 \text{ for } t \not= 0$

      Otherwise, with memory.

  • Invertibility of LTI System

    A system $S$ is invertible if and only if there exists an inverse system $S^{-1}$ such that $S^{-1}$ is an identity system.

    

    $h*h_1 = \delta$

  • Causality Property

    DTS: $h(n) = 0 ,\space n < 0$

    CTS: $h(t) = 0, \space t < 0$

  • Stability Property

    $\sum_{k=\infin}^{\infin} |h(k)| < \infin$, for DTS

    $\int_{-\infin}^{\infin} |h(t)| < \infin$, for CTS

    • Check Causality and Stability for $h(n) = \frac{1}{2}^n u(n)$ (DTS : Discrete Time Signal)

      $u(n) = 0 , \space n < 0$

      $h(n) = \frac{1}{2}^n \times 0; \space n < 0 \\=0$, Causal;

      $\sum_{k = -\infin}^{\infin} h(k) = \sum_{k = -\infin}^{0} \frac{1}{2}^n u(k) + \sum_{k = 0}^{\infin} \frac{1}{2}^n u(k) = 0 + \frac{1}{1 - \frac{1}{2}} = 2 < \infin$, Stable

• Convolution

  Convolution is a mathematical way of “combining two signals to form a third signal”

  $*$ is the convolution sign.

  • Characterizing a LTI System

    

    $y(n) = x(n)*h(n) = h(n)*x(n)$

    $y(t) = x(t)*h(t) = h(t)*x(t)$

    DTS(Discrete Time Signal):

    $y(n) = x(n)*h(n) \\ = h(n)*x(n) \\= \sum_{k=-\infin}^{\infin} x(k) h(n-k) \\= \sum_{k=-\infin}^{\infin} x(n-k) h(n)$

    

    

    CTS (Continuous Time Signal):

    $y(t) = h(t)*x(t) = x(t)*h(t)\\ = \int_{0}^t x(\lambda) h(t-\lambda)d\lambda = \int_{0}^t h(\lambda)x(t-\lambda) d\lambda$

    Steps:

    • Folding: $h(\lambda) = h(-\lambda)$
    • Shifting: $h(-\lambda+t) \rarr h(t-\lambda)$
    • Multiplication: $x(\lambda), h(t-\lambda)$
    • Integrating: $\int_{t_1}^{t_2} x(\lambda)*h(t-\lambda)$

    

    Laplace Transform

    Laplace transform is an integral transformation of a function $f(t)$ from time domain into the frequency domain.

    $f(t) \rarr F(s)$ $s = a+jw$, real and imaginary part

    $\text{Laplace of f(t), } \mathscr{L} [f(t)] = F(s) =\int_0^{\infin} f(t)e^{-st} dt$

    $y(t) = x(t)*h(t) \text{ convulotion in time domain} \\ Y(s) = X(s)\times H(s)\\\text{ simple multiplication frequency domain}$

    $\mathscr{L} [u(t)] = \int_0^{\infin} u(t) e^{-st} dt = \frac{1}{s}$

    $\mathscr{L}[coswt] = \int_0^{\infin} \frac{e^{jwt} + e^{-jwt}}{2} e^{-st} dt = \frac{s}{s^2 + w^2}$

    $\mathscr{L} [sinwt] = \int_0^{\infin} \frac{1}{2j}(e^{jwt} - e^{-jwt}) e^{-st} dt = \frac{w}{s^2 + w^2}$

    $\mathscr{L[tu(t)]} = \int_0^{\infin} t u(t) e^{-st} = t \int_0^{\infin} e^{-st} - \int_0^{\infin} \frac{dt}{dt} \int_0^{\infin} e^{-st} = \frac{1}{s^2}$

    Laplace transform condition/Dirichlet’s conditions

    • The function must be integrable in the given interval of time

      $\int_{-\infin}^{\infin} f(t) dt < \infin$

    • The function should have finite number of maxima and minima.

      Example: Noise signal doesn’t follow this condition

    • There must be finite number of discontinuities in the signal in the given interval of time.

  Properties

  Linearity Property

  If $x_1(t), x_2(t)$ are two signal, then the will follow the superposition theorem.

  $a_1x_1(t) + a_2 x_2(t) \rarr a_1X_1(s) + a_2 X_2(s)$

  $Z_1(t) = a_1x_1(t) + a_2 x_2(t)$

  $X_1(s) = \mathscr{L} [x_1(t)] = \int_{-\infin}^{\infin} x_1(t) e^{-st} dt \\$$X_2(s) = \mathscr{L} [x_2(t)] = \int_{-\infin}^{\infin} x_2(t) e^{-st} dt$

  $Z_1(s) = \mathscr{L} [Z_1(t)] = a_1X_1(s) + a_2X_2(s)$

  Time Shifting Property

  $x(t) \lrarr X(s)$

  $x(t-t_0) \lrarr e^{-st_0} X(s)$

  Time Scaling Property

  $x(at) \lrarr \frac{1}{|a|} X(\frac{s}{a})$

  Freequency Property

  $X(s-s_0) \rarr e^{s_0t} x(t)$

  Time Reversal

  $x(t) \lrarr X(s) \\ x(-t) \lrarr -X(-s)$

  Differentiation & Integration Property

  $\frac{dx(t)}{dt} \lrarr SX(S)$

  $\int x(t) dt \lrarr \frac{X(s)}{s}$

  Multiplication and Convolution Property

  $x_1(t), x_2(t)$

  $x_1(t)*x_2(t) \lrarr X_1(s) \times X_2(s)$

  $x_1(t) \times x_2(t) \lrarr X_1(s) * X_2(s)$

Fourier Series

Fourier Series is a sum that represents a “periodic function” as a sum of $sine$ and $cosine$ waves in terms of their harmonics.

$x(t) = x(x \pm T)$, T : Time period, $x(t)$ is periodic.

Frequency, $\tt f = no \space of \space cycle/sec$ / rate of change

Harmonics

3rd Harmonics will dominant 2nd one because it’s magnitude is greater than the 2nd one. (9 > 5)

Dirichlet conditions of Fourier Series

• Function $f(t)$ is single valued everywhere.

  $f(x) = x^2;$

  For every $x$ there is only one $f(x) \space or \space y$

• Function has a finite number of discontinuities in any time period.

  

• Function has a finite number of maximum and minimum in any time period.

  

• Signal should be absolutely integrable in any time period.

  $\int_{t_0}^{t_0 + T} |f(t)| dt < \infin$

  

Why $\tt sine \ and \ cosine$ are special to Fourier Series ?

• The Fourier series uses cosine and sine functions because they form a complete set of orthogonal functions, meaning they are independent and can be used to represent any periodic function

Trigonometric Fourier Series

$f(t) = a_0 + \sum_{n=1} ^{\infin} a_ncosnw_0t + b_nsinnw_0t ... (i)$

$a_0, a_n, b_n = \text{fourier coneffecient}$

$a_0 = \frac{1}{T} \int_0^{T} f(t) dt = \text{average of function}$

Multiplying $cosmw_0t$ and integrating in $(i),$

$\int_0^T f(t) cosmw_0tdt = \int_0^T [a_0 + \sum_{n=1} ^{\infin} a_ncosnw_0t + b_nsinnw_0t] cosmw_0t dt$

$= \int_0^ta_0cosmw_0tdt + \sum_{n=1}^{\infin} ( \int_0^ta_ncosmw_0t \cdot cosnw_0t dt+ \int_0^ta_ncosmwt \cdot sinnw_0t) dt$

$=0 + \sum_{n=1}^{\infin} \int_0^ta_ncosmw_0t \cdot cosnw_0t dt+ 0$

$\tt Now, n=m,$

Multiplying $sinmw_0t$ and integrating in $(i)$,

$b_n = \frac{2}{T} \int_0^T f(t) sinnw_0t dt$

Fourier Series Expansion

$f(t) = \begin{cases} 1 & 0 \le t \le 1 \\ 0 & 1 \le t \le 2 \end{cases}$