signal formulaes

Here is the exhaustive Master Formula Sheet for ECE 2107. Every equation, derivation step, condition, and circuit model from your lectures, textbook, and PYQs has been extracted and rigorously formatted to optimize your A+ preparation.

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Instructor 1: Transform Techniques (Fourier, Laplace, Z-Transform, FFT)

1. Fourier Series

[Trigonometric Fourier Series Representation]

Formula: $f(t)=\frac{a_0}{2}+{\sum }_{n=1}^{\infty }{a}_n\cos (n{\omega }_{0}t)+{\sum }_{n=1}^{\infty }{b}_n\sin (n{\omega }_{0}t)$ Concept: Expresses a periodic, non-sinusoidal signal as a linear combination of fundamental and harmonic sine and cosine waves. Applies to any periodic signal satisfying Dirichlet conditions. [PYQ: 2021, 2020] [Heavily Tested] Symbols:

• $f(t)$ = Periodic time-domain signal [unit depends on physical quantity, e.g., Volts or Amps]
• $a_0$ = DC component/average value coefficient [unit same as $f(t)$]
• $a_n$ = Cosine harmonic coefficient [unit same as $f(t)$]
• $b_n$ = Sine harmonic coefficient [unit same as $f(t)$]
• $n$ = Integer harmonic number [unitless]
• ${\omega }_0$ = Fundamental angular frequency [rad/s]
• $t$ = Time [seconds]

[DC Component / Average Value ($a_0$)]

Formula: $a_0=\frac{2}{T_0}{\int }_0^{T_0}f(t),dt$ Concept: Calculates the constant baseline or DC offset of a periodic signal over one fundamental period. Symbols:

• $a_0$ = DC component coefficient [unit same as $f(t)$]
• $T_0$ = Fundamental time period [seconds]
• $f(t)$ = Periodic time-domain signal [unit depends on signal]
• $t$ = Time [seconds]

[Cosine Harmonic Coefficient ($a_n$)]

Formula: $a_n=\frac{2}{T_0}{\int }_0^{T_0}f(t)\cos (n{\omega }_{0}t),dt$ Concept: Calculates the amplitude of the $n$-th cosine harmonic in a periodic signal. For odd functions, this mathematically evaluates to zero. [PYQ: 2017, 2018] Symbols:

• $a_n$ = Amplitude of the $n$-th cosine term [unit same as $f(t)$]
• $T_0$ = Fundamental time period [seconds]
• $f(t)$ = Periodic time-domain signal [unit depends on signal]
• $n$ = Harmonic integer index [unitless]
• ${\omega }_0$ = Fundamental angular frequency [rad/s]
• $t$ = Time [seconds]

[Sine Harmonic Coefficient ($b_n$)]

Formula: $b_n=\frac{2}{T_0}{\int }_0^{T_0}f(t)\sin (n{\omega }_{0}t),dt$ Concept: Calculates the amplitude of the $n$-th sine harmonic in a periodic signal. For even functions, this mathematically evaluates to zero. [PYQ: 2017, 2018] Symbols:

• $b_n$ = Amplitude of the $n$-th sine term [unit same as $f(t)$]
• $T_0$ = Fundamental time period [seconds]
• $f(t)$ = Periodic time-domain signal [unit depends on signal]
• $n$ = Harmonic integer index [unitless]
• ${\omega }_0$ = Fundamental angular frequency [rad/s]
• $t$ = Time [seconds]

[Complex / Exponential Fourier Series Representation]

Formula: $f(t)={\sum }_{n=-\infty }^{\infty }{C}_n{e}^{jn{\omega }_{0}t}$ Concept: The compact representation of a periodic signal using complex exponentials, bridging the gap between Fourier Series and Fourier Transform. [PYQ: 2021, 2018] [Heavily Tested] Symbols:

• $f(t)$ = Periodic time-domain signal [unit depends on signal]
• $C_n$ = Complex Fourier coefficient [unit same as $f(t)$]
• $j$ = Imaginary unit ($\sqrt{-1}$) [unitless]
• $n$ = Integer harmonic number (positive and negative) [unitless]
• ${\omega }_0$ = Fundamental angular frequency [rad/s]
• $t$ = Time [seconds]

[Complex Fourier Coefficient ($C_n$)]

Formula: $C_n=\frac{1}{T_0}{\int }_{-T_0/2}^{T_0/2}f(t)e^{-jn{\omega }_{0}t},dt$ Concept: Calculates both the magnitude and phase of the $n$-th frequency component in the complex exponential Fourier series. Symbols:

• $C_n$ = Complex Fourier coefficient [unit same as $f(t)$]
• $T_0$ = Fundamental time period [seconds]
• $f(t)$ = Periodic time-domain signal [unit depends on signal]
• $j$ = Imaginary unit [unitless]
• $n$ = Harmonic integer index [unitless]
• ${\omega }_0$ = Fundamental angular frequency [rad/s]
• $t$ = Time [seconds]

[Relationship Between Complex and Trigonometric Coefficients]

Formula: $C_n=\frac{a_n-j{b}_n}{2}$ Concept: Maps the coefficients of the trigonometric Fourier series directly to the complex exponential Fourier series (for $n>0$). Symbols:

• $C_n$ = Complex Fourier coefficient [unit same as signal]
• $a_n$ = Cosine harmonic coefficient [unit same as signal]
• $b_n$ = Sine harmonic coefficient [unit same as signal]
• $j$ = Imaginary unit [unitless]

[Parseval’s Identity for Fourier Series]

Formula: $\frac{1}{T}{\int }_{-T/2}^{T/2}[f(t){]}^2,dt={\left(\frac{a_0}{2}\right)}^2+\frac{1}{2}{\sum }_{n=1}^{\infty }(a_n^2+b_n^2)$ Concept: Proves that the total average power of a periodic signal in the time domain is equal to the sum of the average powers of its individual harmonic components. [PYQ: 2020, 2019] [Heavily Tested] Symbols:

• $T$ = Fundamental time period [seconds]
• $f(t)$ = Periodic signal [Volts or Amps]
• $a_0,a_n,b_n$ = Trigonometric Fourier coefficients [Volts or Amps]

[Mean Square Error (MSE) of Fourier Approximation]

Formula: $E_N=\frac{1}{2l}{\int }_{-l}^l(f(x)-S_N(x){)}^2,dx$ Concept: Calculates the error energy when a signal $f(x)$ is approximated by a finite number $N$ of Fourier harmonic terms $S_N(x)$. Symbols:

• $E_N$ = Mean Square Error [unit squared]
• $2l$ = Fundamental period $T$ [seconds or spatial units]
• $f(x)$ = Original signal [unit depends on signal]
• $S_N(x)$ = $N$-th partial sum of Fourier series [unit same as $f(x)$]
• $x$ = Independent variable (usually time) [seconds]

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2. Continuous-Time Fourier Transform

[Fourier Transform Definition]

Formula: $F(j\omega )={\int }_{-\infty }^{\infty }f(t)e^{-j\omega t},dt$ Concept: Converts an aperiodic time-domain signal into a continuous frequency-domain spectrum, revealing frequency content density. [PYQ: 2025, 2024, 2016, 2015] [Heavily Tested] Symbols:

• $F(j\omega )$ = Frequency-domain spectrum [signal unit $\cdot$ seconds]
• $f(t)$ = Time-domain signal [unit depends on signal]
• $j$ = Imaginary unit [unitless]
• $\omega$ = Angular frequency [rad/s]
• $t$ = Time [seconds]

[Inverse Fourier Transform]

Formula: $f(t)=\frac{1}{2\pi }{\int }_{-\infty }^{\infty }F(j\omega )e^{j\omega t},d\omega$ Concept: Reconstructs the exact time-domain signal from its continuous frequency-domain spectrum. [PYQ: 2025, 2023] Symbols:

• $f(t)$ = Time-domain signal [unit depends on signal]
• $F(j\omega )$ = Frequency-domain spectrum [signal unit $\cdot$ seconds]
• $j$ = Imaginary unit [unitless]
• $\omega$ = Angular frequency [rad/s]
• $t$ = Time [seconds]

[Rayleigh’s Energy Theorem (Parseval’s for FT)]

Formula: $E={\int }_{-\infty }^{\infty }|f(t){|}^2,dt=\frac{1}{2\pi }{\int }_{-\infty }^{\infty }|F(j\omega ){|}^2,d\omega$ Concept: States that total energy computed by integrating the signal squared over all time equals the energy computed by integrating the energy spectral density over all frequencies. [PYQ: 2022, 2016] [Heavily Tested] Symbols:

• $E$ = Total signal energy [Joules, or signal unit${}^2\cdot$ seconds]
• $f(t)$ = Time-domain signal [unit depends on signal]
• $F(j\omega )$ = Fourier transform spectrum [signal unit $\cdot$ seconds]
• $\omega$ = Angular frequency [rad/s]
• $t$ = Time [seconds]

[Time Shifting Property]

Formula: $f(t-t_0)\leftrightarrow F(j\omega )e^{-j\omega t_0}$ Concept: A delay in the time domain corresponds to a linear phase shift in the frequency domain without altering the magnitude spectrum. [PYQ: 2023] Symbols:

• $f(t-t_0)$ = Time-delayed signal [unit depends on signal]
• $t_0$ = Time delay constant [seconds]
• $F(j\omega )$ = Fourier transform of original signal [signal unit $\cdot$ seconds]
• $\omega$ = Angular frequency [rad/s]

[Frequency Shifting (Modulation) Property]

Formula: $f(t)e^{j{\omega }_{0}t}\leftrightarrow F(j(\omega -{\omega }_0))$ Concept: Multiplying a signal by a complex exponential in time shifts its entire frequency spectrum by ${\omega }_0$. Core foundation of amplitude modulation (AM). [PYQ: 2021] Symbols:

• $f(t)$ = Baseband time-domain signal [unit depends on signal]
• ${\omega }_0$ = Carrier/shifting angular frequency [rad/s]
• $F(j(\omega -{\omega }_0))$ = Frequency-shifted spectrum [signal unit $\cdot$ seconds]
• $t$ = Time [seconds]

[Time Scaling Property]

Formula: $f(at)\leftrightarrow \frac{1}{|a|}F\left(j\frac{\omega }{a}\right)$ Concept: Compressing a signal in time ($a>1$) expands its frequency spectrum, demonstrating the inverse relationship between time duration and bandwidth. [PYQ: 2021] Symbols:

• $f(at)$ = Time-scaled signal [unit depends on signal]
• $a$ = Real scaling constant [unitless]
• $F$ = Fourier transform operator
• $\omega$ = Angular frequency [rad/s]

[Time Differentiation Property]

Formula: $\frac{d^n}{d{t}^n}f(t)\leftrightarrow (j\omega {)}^{n}F(j\omega )$ Concept: Taking the derivative of a signal in the time domain equates to multiplying its Fourier transform by $j\omega$. Useful for solving differential equations. [PYQ: 2021] Symbols:

• $n$ = Order of differentiation [integer]
• $f(t)$ = Time-domain signal [unit depends on signal]
• $\omega$ = Angular frequency [rad/s]
• $F(j\omega )$ = Fourier transform [signal unit $\cdot$ seconds]

[Time Convolution Property]

Formula: $x(t)\ast h(t)\leftrightarrow X(j\omega )H(j\omega )$ Concept: Convolution in the time domain simplifies to direct algebraic multiplication in the frequency domain. [PYQ: 2022, 2018] [Heavily Tested] Symbols:

• $x(t)$ = Input signal [unit depends on signal]
• $h(t)$ = Impulse response [unit depends on system]
• $\ast$ = Convolution operator
• $X(j\omega )$ = FT of input signal
• $H(j\omega )$ = FT of impulse response (Transfer Function)

[Fourier Transform of Normalized Gaussian Pulse]

Formula: $e^{-\pi t^2}\leftrightarrow e^{-\pi f^2}$ (or $e^{-{\omega }^2/4\pi }$) Concept: The unique mathematical pulse that perfectly maintains its shape across both the time and frequency domains. [PYQ: 2024, 2018, 2017] [Heavily Tested] Symbols:

• $e$ = Euler’s number (approx 2.718) [unitless]
• $t$ = Time [seconds]
• $f$ = Frequency ($\omega /2\pi$) [Hz]

[Fourier Transform of Rectangular Pulse (Gate)]

Formula: $\text{rect}\left(\frac{t}{T}\right)\leftrightarrow T\text{sinc}\left(\frac{\omega T}{2}\right)$ Concept: A sharp rectangular pulse in time generates a highly oscillatory ‘sinc’ function spectrum, proving that instantaneous changes require infinite bandwidth. Symbols:

• $\text{rect}$ = Rectangular pulse function (amplitude 1 from $-T/2$ to $T/2$) [unitless]
• $t$ = Time [seconds]
• $T$ = Total pulse width [seconds]
• $\text{sinc}(x)$ = Defined as $\frac{\sin (x)}{x}$ [unitless]
• $\omega$ = Angular frequency [rad/s]

[Fourier Transform of Unit Impulse (Dirac Delta)]

Formula: $\delta (t)\leftrightarrow 1$ Concept: An infinitely narrow impulse contains absolutely all frequencies uniformly. Symbols:

• $\delta (t)$ = Dirac delta function [1/seconds]
• $1$ = Uniform frequency amplitude [unitless]

[Fourier Transform of Signum Function]

Formula: $\text{sgn}(t)\leftrightarrow \frac{2}{j\omega }$ Concept: The frequency spectrum of a function that flips from -1 to +1 at $t=0$. [PYQ: 2022, 2019] [Heavily Tested] Symbols:

• $\text{sgn}(t)$ = Signum function [unitless]
• $j$ = Imaginary unit [unitless]
• $\omega$ = Angular frequency [rad/s]

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3. Laplace Transform

[Bilateral Laplace Transform]

Formula: $F(s)={\int }_{-\infty }^{\infty }f(t)e^{-st},dt$ Concept: Evaluates systems covering all time (past and future), mapping signals to the complex s-plane. Symbols:

• $F(s)$ = Laplace transform [signal unit $\cdot$ seconds]
• $f(t)$ = Time-domain signal [unit depends on signal]
• $s$ = Complex frequency variable ($\sigma +j\omega$) [Complex rad/s]
• $t$ = Time [seconds]

[Unilateral Laplace Transform]

Formula: $F(s)={\int }_0^{\infty }f(t)e^{-st},dt$ Concept: The standard form used in engineering for causal signals ($t\ge 0$) to solve differential equations with initial conditions. [PYQ: 2019, 2017, 2016, 2015] [Heavily Tested] Symbols:

• $F(s)$ = Laplace transform [signal unit $\cdot$ seconds]
• $f(t)$ = Time-domain signal [unit depends on signal]
• $s$ = Complex frequency variable ($\sigma +j\omega$) [Complex rad/s]
• $t$ = Time [seconds]

[Inverse Laplace Transform]

Formula: $f(t)=\frac{1}{2\pi j}{\int }_{\sigma -j\infty }^{\sigma +j\infty }F(s)e^{st},ds$ Concept: The contour integration used to convert a complex frequency-domain function back to the time domain. Typically solved using Partial Fraction Expansion instead. [PYQ: 2019, 2017] Symbols:

• $f(t)$ = Time-domain signal [unit depends on signal]
• $F(s)$ = Laplace domain function [signal unit $\cdot$ seconds]
• $s$ = Complex frequency variable [Complex rad/s]
• $\sigma$ = Real part of $s$ defining the vertical integration line [Neper/s]

[Initial Value Theorem]

Formula: ${\lim }_{t\to 0^{+}}f(t)={\lim }_{s\to \infty }sF(s)$ Concept: Determines the instantaneous starting value of a time-domain signal directly from its s-domain equation, without performing inverse Laplace. [PYQ: 2023, 2021, 2019, 2016] [Heavily Tested] Symbols:

• $t$ = Time [seconds]
• $f(t)$ = Time domain signal [unit depends on signal]
• $s$ = Complex frequency [Complex rad/s]
• $F(s)$ = Laplace transform of $f(t)$

[Final Value Theorem]

Formula: ${\lim }_{t\to \infty }f(t)={\lim }_{s\to 0}sF(s)$ Concept: Evaluates the steady-state long-term DC behavior of a stable system directly from the s-domain. Condition: Roots of denominator of $sF(s)$ must lie in the left half s-plane. [PYQ: 2023, 2021, 2019, 2016] [Heavily Tested] Symbols:

• $t$ = Time [seconds]
• $f(t)$ = Time domain signal [unit depends on signal]
• $s$ = Complex frequency [Complex rad/s]
• $F(s)$ = Laplace transform of $f(t)$

[Time Differentiation Property (with Initial Conditions)]

Formula: \mathcal{L}\left{\frac{df(t)}{dt}\right} = sF(s) - f(0^-) Concept: Translates a time derivative into an algebraic multiplication by $s$, subtracting the initial condition. Core to solving circuit transients. Symbols:

• $\mathcal{L}$ = Laplace transform operator
• $f(t)$ = Time-domain signal
• $s$ = Complex frequency [Complex rad/s]
• $F(s)$ = Laplace transform of $f(t)$
• $f(0^{-})$ = Initial condition just before $t=0$

[S-Domain Differentiation Property]

Formula: $\mathcal{L}-tf(t)=\frac{d}{ds}F(s)$ Concept: Multiplying a signal by time $t$ corresponds to differentiating its Laplace transform with respect to $s$. [PYQ: 2023] Symbols:

• $\mathcal{L}$ = Laplace transform operator
• $t$ = Time [seconds]
• $f(t)$ = Time-domain signal
• $s$ = Complex frequency [Complex rad/s]
• $F(s)$ = Laplace transform

[S-Domain Equivalent of Inductor (with Initial Current)]

Formula: $V_L(s)=LsI(s)-Li(0^{-})$ Concept: Transforms an inductor into the s-domain, accounting for the energy stored in its magnetic field prior to $t=0$ as an independent voltage source. [PYQ: 2024, 2021, 2018, 2017] [Heavily Tested] Symbols:

• $V_L(s)$ = Inductor voltage in s-domain [Volts $\cdot$ seconds]
• $L$ = Inductance [Henries]
• $s$ = Complex frequency [Complex rad/s]
• $I(s)$ = Inductor current in s-domain [Amps $\cdot$ seconds]
• $i(0^{-})$ = Initial current flowing through inductor at $t=0$ [Amps]

[S-Domain Equivalent of Capacitor (with Initial Voltage)]

Formula: $V_C(s)=\frac{1}{Cs}I(s)+\frac{v_c(0^{-})}{s}$ Concept: Transforms a capacitor into the s-domain, accounting for the energy stored in its electric field prior to $t=0$ as a step voltage source. [PYQ: 2019, 2018, 2015] [Heavily Tested] Symbols:

• $V_C(s)$ = Capacitor voltage in s-domain [Volts $\cdot$ seconds]
• $C$ = Capacitance [Farads]
• $s$ = Complex frequency [Complex rad/s]
• $I(s)$ = Capacitor current in s-domain [Amps $\cdot$ seconds]
• $v_c(0^{-})$ = Initial voltage across capacitor at $t=0$ [Volts]

[Transfer Function (S-Domain)]

Formula: $H(s)=\frac{Y(s)}{X(s)}=\frac{(s-z_1)(s-z_2)\dots }{(s-p_1)(s-p_2)\dots }$ Concept: The ratio of the output Laplace transform to the input Laplace transform, assuming zero initial conditions. Determines system poles ($p$) and zeros ($z$). [PYQ: 2023, 2020, 2019] [Heavily Tested] Symbols:

• $H(s)$ = Transfer function [unitless or system specific]
• $Y(s)$ = Output signal in s-domain
• $X(s)$ = Input signal in s-domain
• $z_k$ = Complex zero frequencies [Complex rad/s]
• $p_k$ = Complex pole frequencies [Complex rad/s]

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4. Z-Transform

[Bilateral Z-Transform]

Formula: $X(z)={\sum }_{n=-\infty }^{\infty }x(n)z^{-n}$ Concept: Converts a discrete-time sequence into the complex z-domain. Exists only within a specific Region of Convergence (ROC). [PYQ: 2022, 2018, 2016] [Heavily Tested] Symbols:

• $X(z)$ = Z-transform function
• $x(n)$ = Discrete-time sequence [unit depends on signal]
• $z$ = Complex variable ($r{e}^{j\omega }$) [unitless]
• $n$ = Discrete time index [integer]

[Unilateral Z-Transform]

Formula: $X(z)={\sum }_{n=0}^{\infty }x(n)z^{-n}$ Concept: Computes the Z-transform strictly for causal signals ($n\ge 0$), heavily used for solving difference equations with initial conditions. [PYQ: 2020] Symbols:

• $X(z)$ = Z-transform function
• $x(n)$ = Discrete-time sequence
• $z$ = Complex variable [unitless]
• $n$ = Discrete time index [integer]

[Inverse Z-Transform via Cauchy’s Residue Theorem]

Formula: $x(n)=\frac{1}{2\pi j}{\oint }_{C}X(z)z^{n-1},dz=\sum \left[\text{Residues of }X(z)z^{n-1}\text{ at poles}\right]$ Concept: The formal contour integral definition of the inverse Z-transform. [PYQ: 2019] Symbols:

• $x(n)$ = Reconstructed discrete-time sequence
• $X(z)$ = Z-domain function
• $z$ = Complex variable
• $C$ = Counter-clockwise closed contour in the ROC

[Time Shifting (Delay) Property]

Formula: $x(n-k)\leftrightarrow z^{-k}X(z)$ Concept: Delaying a discrete sequence by $k$ samples corresponds to multiplying its Z-transform by $z^{-k}$. Core to converting difference equations. [PYQ: 2022, 2018, 2017] [Heavily Tested] Symbols:

• $x(n-k)$ = Delayed discrete sequence
• $k$ = Integer delay amount [samples]
• $z^{-k}$ = Delay operator in z-domain
• $X(z)$ = Z-transform of original signal

[Initial Value Theorem (Z-Domain)]

Formula: $x(0)={\lim }_{z\to \infty }X(z)$ Concept: Finds the very first sample value of a causal sequence directly from the Z-domain equation. [PYQ: 2025, 2022, 2020, 2019] [Heavily Tested] Symbols:

• $x(0)$ = Value of the sequence at index $n=0$
• $z$ = Complex variable
• $X(z)$ = Z-transform of causal sequence

[Final Value Theorem (Z-Domain)]

Formula: $x(\infty )={\lim }_{z\to 1}(1-z^{-1})X(z)={\lim }_{z\to 1}\frac{z-1}{z}X(z)$ Concept: Calculates the steady-state final value of a stable discrete sequence. Condition: All poles of $(1-z^{-1})X(z)$ must be inside the unit circle. [PYQ: 2025, 2024, 2020, 2019, 2015] [Heavily Tested] Symbols:

• $x(\infty )$ = Steady-state limit of sequence as $n\to \infty$
• $z$ = Complex variable
• $X(z)$ = Z-transform of sequence

[Discrete Convolution Property]

Formula: $x_1(n)\ast x_2(n)\leftrightarrow X_1(z)X_2(z)$ Concept: Time-domain convolution of discrete sequences maps exactly to multiplication in the z-domain. [PYQ: 2024, 2022, 2018, 2016] [Heavily Tested] Symbols:

• $x_1(n),x_2(n)$ = Discrete sequences
• $\ast$ = Discrete convolution sum operator
• $X_1(z),X_2(z)$ = Z-transforms of the respective sequences

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5. DFT & FFT

[Discrete Fourier Transform (DFT)]

Formula: $X(k)={\sum }_{n=0}^{N-1}x(n)W_N^{nk}\text{for }0\le k\le N-1$ Concept: Converts a finite N-point discrete-time sequence into an N-point discrete frequency spectrum. Computed inefficiently in $O(N^2)$ time. Symbols:

• $X(k)$ = DFT sequence (frequency domain)
• $x(n)$ = Input discrete sequence (time domain)
• $N$ = Total number of sample points
• $n$ = Time index integer
• $k$ = Frequency index integer
• $W_N$ = Twiddle / phase factor

[Inverse Discrete Fourier Transform (IDFT)]

Formula: $x(n)=\frac{1}{N}{\sum }_{k=0}^{N-1}X(k)W_N^{-nk}\text{for }0\le n\le N-1$ Concept: Reconstructs the finite discrete-time sequence exactly from its DFT bins. [PYQ: 2024] Symbols:

• $x(n)$ = Reconstructed time sequence
• $N$ = Total number of points
• $X(k)$ = DFT sequence
• $W_N^{-nk}$ = Inverse phase factor

[Twiddle / Phase Factor ($W_N$)]

Formula: $W_N=e^{-j\frac{2\pi }{N}}$ Concept: The complex exponential root of unity that drives the rotations in DFT and FFT algorithms. Exhibits symmetry ($W_N^{k+N/2}=-W_N^k$) and periodicity ($W_N^{k+N}=W_N^k$). Symbols:

• $W_N$ = Phase factor [unitless]
• $e$ = Euler’s number
• $j$ = Imaginary unit
• $N$ = Total number of points in transform

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Instructor 2: Signals, Systems, Filters and Network Theory

1. Signal Fundamentals

[Even Part of a Signal]

Formula: $x_e(t)=\frac{x(t)+x(-t)}{2}$ Concept: Extracts the perfectly symmetric portion of any arbitrary signal. [PYQ: 2024, 2020, 2019, 2018, 2015] [Heavily Tested] Symbols:

• $x_e(t)$ = Even component
• $x(t)$ = Original signal
• $x(-t)$ = Time-reversed signal

[Odd Part of a Signal]

Formula: $x_o(t)=\frac{x(t)-x(-t)}{2}$ Concept: Extracts the strictly anti-symmetric portion of any arbitrary signal. [PYQ: 2024, 2020, 2019, 2018, 2015] [Heavily Tested] Symbols:

• $x_o(t)$ = Odd component
• $x(t)$ = Original signal
• $x(-t)$ = Time-reversed signal

[Total Energy of Continuous-Time Signal]

Formula: $E={\int }_{-\infty }^{\infty }|x(t){|}^2,dt$ Concept: Calculates the total dissipated energy of a transient or aperiodic signal across infinite time. Signal is an “Energy Signal” if $0<E<\infty$. [PYQ: 2025, 2023, 2022, 2019, 2018, 2017, 2016, 2015] [Heavily Tested] Symbols:

• $E$ = Total energy [Joules]
• $x(t)$ = Signal amplitude [Volts/Amps]
• $t$ = Time [seconds]

[Average Power of Continuous-Time Signal]

Formula: $P={\lim }_{T\to \infty }\frac{1}{T}{\int }_{-T/2}^{T/2}|x(t){|}^2,dt$ Concept: Calculates the average power dissipation for infinite-duration or periodic signals. Signal is a “Power Signal” if $0<P<\infty$. [PYQ: 2025, 2023, 2022, 2019, 2018, 2017, 2016, 2015] [Heavily Tested] Symbols:

• $P$ = Average power [Watts]
• $T$ = Observation time window [seconds]
• $x(t)$ = Signal amplitude [Volts/Amps]

[Energy of Discrete-Time Signal]

Formula: $E={\sum }_{n=-\infty }^{\infty }|x(n){|}^2$ Concept: The discrete summation equivalent of total signal energy. Symbols:

• $E$ = Total energy [Joules]
• $x(n)$ = Discrete sequence amplitude
• $n$ = Integer index

[Periodicity Condition (Composite Continuous Signals)]

Formula: $\frac{T_1}{T_2}=\text{Rational Number}$ Concept: For a signal composed of multiple sinusoids (with individual periods $T_1,T_2$), the overall signal is periodic only if the ratio of their periods is a rational fraction. The fundamental period is the LCM of the periods. [PYQ: 2025, 2019, 2017, 2016] [Heavily Tested] Symbols:

• $T_1$ = Fundamental period of first sub-signal [seconds]
• $T_2$ = Fundamental period of second sub-signal [seconds]

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2. System Properties & Convolution

[Continuous-Time Convolution Integral]

Formula: $y(t)=x(t)\ast h(t)={\int }_{-\infty }^{\infty }x(\tau )h(t-\tau ),d\tau$ Concept: The fundamental mathematical operation calculating the exact time-domain output $y(t)$ of an LTI system given an input $x(t)$ sliding across an impulse response $h(t)$. [PYQ: 2023, 2021, 2017, 2015] [Heavily Tested] Symbols:

• $y(t)$ = System output [unit depends on system]
• $x(t)$ = System input signal
• $h(t)$ = System impulse response
• $\tau$ = Integration dummy time variable [seconds]
• $t$ = Actual time [seconds]

[Discrete-Time Convolution Sum]

Formula: $y(n)=x(n)\ast h(n)={\sum }_{k=-\infty }^{\infty }x(k)h(n-k)$ Concept: The discrete counterpart of the convolution integral, aggregating the weighted and shifted impulse responses triggered by each input sample. [PYQ: 2022, 2021, 2020, 2018, 2017, 2015] [Heavily Tested] Symbols:

• $y(n)$ = Output sequence
• $x(n)$ = Input sequence
• $h(n)$ = System impulse response sequence
• $k$ = Dummy integer shift variable
• $n$ = Discrete time index

[BIBO Stability Condition (Continuous LTI)]

Formula: ${\int }_{-\infty }^{\infty }|h(t)|,dt<\infty$ Concept: A continuous LTI system is Bounded-Input Bounded-Output (BIBO) stable if and only if its impulse response is absolutely integrable. [PYQ: 2017, 2016, 2015] [Heavily Tested] Symbols:

• $h(t)$ = System impulse response
• $t$ = Time [seconds]
• $\infty$ = Infinity boundary

[BIBO Stability Condition (Discrete LTI)]

Formula: ${\sum }_{n=-\infty }^{\infty }|h(n)|<\infty$ Concept: A discrete LTI system is BIBO stable if and only if its impulse response sequence is absolutely summable. Symbols:

• $h(n)$ = Discrete impulse response sequence
• $n$ = Integer time index

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3. State Space Representation

[Continuous-Time State Equation]

Formula: $\dot{X}(t)=AX(t)+BU(t)$ Concept: A set of first-order differential equations in matrix form that mathematically track the internal energy states of a dynamic system over time. [PYQ: 2024, 2021, 2019, 2018, 2017, 2016] [Heavily Tested] Symbols:

• $\dot{X}(t)$ = Time derivative of state vector ($n\times 1$)
• $X(t)$ = State vector containing state variables ($n\times 1$)
• $A$ = System/State matrix ($n\times n$)
• $B$ = Input matrix ($n\times m$)
• $U(t)$ = Input vector ($m\times 1$)

[Continuous-Time Output Equation]

Formula: $Y(t)=CX(t)+DU(t)$ Concept: Algebraic matrix equation mapping the internal dynamic state variables $X(t)$ and direct inputs $U(t)$ to the observable outputs $Y(t)$. Symbols:

• $Y(t)$ = Output vector ($p\times 1$)
• $C$ = Output matrix ($p\times n$)
• $X(t)$ = State vector ($n\times 1$)
• $D$ = Direct transmission/Feedforward matrix ($p\times m$)
• $U(t)$ = Input vector ($m\times 1$)

[State Transition Matrix (Infinite Series Method)]

Formula: $\varphi (t)=e^{At}=I+At+\frac{(At{)}^2}{2!}+\frac{(At{)}^3}{3!}+\dots$ Concept: The matrix exponential that dictates the unforced (zero-input) natural response decay/oscillation of the system’s states over time. Symbols:

• $\varphi (t)$ = State transition matrix ($n\times n$)
• $e^{At}$ = Matrix exponential
• $A$ = System matrix ($n\times n$)
• $I$ = Identity matrix ($n\times n$)
• $t$ = Time [seconds]

[State Transition Matrix (Laplace Method)]

Formula: $\varphi (t)={\mathcal{L}}^{-1}(sI-A{)}^{-1}$ Concept: The practical algebraic method for finding $e^{At}$ by calculating the inverse Laplace transform of the resolvent matrix. Symbols:

• $\varphi (t)$ = State transition matrix
• ${\mathcal{L}}^{-1}$ = Inverse Laplace operator
• $s$ = Complex frequency variable
• $I$ = Identity matrix
• $A$ = System matrix

[State-Space to Transfer Function Conversion]

Formula: $H(s)=C(sI-A{)}^{-1}B+D$ Concept: Bridges the modern matrix state-space model back to the classical s-domain transfer function used in basic circuit analysis. Symbols:

• $H(s)$ = Transfer function matrix
• $C$ = Output matrix
• $sI-A$ = Resolvent matrix inverse
• $B$ = Input matrix
• $D$ = Feedforward matrix

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4. Sampling & Analog Filters

[Shannon-Nyquist Sampling Theorem Rate]

Formula: $f_s\ge 2f_{max}$ Concept: Establishes the absolute minimum sampling frequency $f_s$ required to perfectly digitize an analog signal without destructive aliasing distortion. [PYQ: 2024, 2023, 2022, 2019, 2018, 2017, 2016, 2015] [Heavily Tested] Symbols:

• $f_s$ = Sampling frequency [Hz or samples/second]
• $f_{max}$ = Highest frequency component present in the analog signal [Hz]

[Butterworth Magnitude Response]

Formula: $|H(j\omega )|=\frac{1}{\sqrt{1+{ϵ}^2{\left(\frac{\omega }{{\omega }_c}\right)}^{2n}}}$ Concept: Mathematical model of a maximally flat low-pass filter. Shows how amplitude rolls off based on the filter order $n$. Symbols:

• $|H(j\omega )|$ = Magnitude of filter transfer function [unitless]
• $ϵ$ = Passband ripple factor [unitless]
• $\omega$ = Operating angular frequency [rad/s]
• ${\omega }_c$ = Cutoff edge frequency [rad/s]
• $n$ = Filter order [integer]

[Butterworth Filter Ripple Factor ($ϵ$)]

Formula: $ϵ=\sqrt{10^{A_p/10}-1}$ Concept: Converts the design specification of passband maximum attenuation $A_p$ (in dB) into the algebraic ripple factor required for the transfer function. [PYQ: 2025, 2023] Symbols:

• $ϵ$ = Ripple factor [unitless]
• $A_p$ = Maximum allowed passband attenuation ripple [Decibels, dB]

[Butterworth Pole Locations]

Formula: $p_k={\omega }_c{e}^{j\left(\frac{\pi }{2}+\frac{(2k-1)\pi }{2n}\right)}\text{for }k=1,2,\dots ,n$ Concept: Dictates exactly where the $n$ stable poles of a Butterworth filter must be placed on the left-half s-plane semicircle to achieve the maximally flat response. Symbols:

• $p_k$ = Complex location of the $k$-th pole
• ${\omega }_c$ = Cutoff frequency (radius of the pole circle) [rad/s]
• $j$ = Imaginary unit
• $n$ = Filter order
• $k$ = Pole integer index

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5. Network Theory (Two-Port Parameters)

[Z-Parameters (Impedance Matrix)]

Formula: $V_1=Z_{11}{I}_1+Z_{12}{I}_2$ $V_2=Z_{21}{I}_1+Z_{22}{I}_2$ Concept: Models a two-port “black box” circuit by expressing the dependent port voltages as a linear combination of the independent port currents (Open-Circuit parameters). [PYQ: 2022] Symbols:

• $V_1,V_2$ = Voltages at port 1 and port 2 [Volts]
• $I_1,I_2$ = Currents entering port 1 and port 2 [Amperes]
• $Z_{11},Z_{12},Z_{21},Z_{22}$ = Impedance parameters [Ohms, $\Omega$]

[Y-Parameters (Admittance Matrix)]

Formula: $I_1=Y_{11}{V}_1+Y_{12}{V}_2$ $I_2=Y_{21}{V}_1+Y_{22}{V}_2$ Concept: Models the circuit by expressing the dependent port currents as a linear combination of the independent port voltages (Short-Circuit parameters). Matrix is the exact inverse of the Z-matrix ($[Y]=[Z{]}^{-1}$). [PYQ: 2025, 2022] [Heavily Tested] Symbols:

• $I_1,I_2$ = Currents entering port 1 and port 2 [Amperes]
• $V_1,V_2$ = Voltages at port 1 and port 2 [Volts]
• $Y_{11},Y_{12},Y_{21},Y_{22}$ = Admittance parameters [Siemens or Mhos, $\mho$]

[ABCD-Parameters (Transmission Matrix)]

Formula: $V_1=A{V}_2-B{I}_2$ $I_1=C{V}_2-D{I}_2$ Concept: Relates the sending-end voltage and current directly to the receiving-end voltage and current. Uniquely optimized for cascading multiple network blocks (matrices multiply directly). Note the minus sign on $I_2$. [PYQ: 2025, 2024, 2022] [Heavily Tested] Symbols:

• $V_1,I_1$ = Sending-end voltage [V] and current [A]
• $V_2$ = Receiving-end voltage [V]
• $I_2$ = Current leaving the receiving end [A]
• $A$ = Reverse voltage ratio [unitless]
• $B$ = Transfer impedance [Ohms, $\Omega$]
• $C$ = Transfer admittance [Siemens, $\mho$]
• $D$ = Reverse current ratio [unitless]

[h-Parameters (Hybrid Matrix)]

Formula: $V_1=h_{11}{I}_1+h_{12}{V}_2$ $I_2=h_{21}{I}_1+h_{22}{V}_2$ Concept: Uses a hybrid mixture of short-circuit and open-circuit measurements. Widely used for modeling transistor circuits (BJT/FET amplifier models). Symbols:

• $V_1,V_2$ = Port voltages [Volts]
• $I_1,I_2$ = Port currents [Amperes]
• $h_{11}$ = Input impedance [Ohms]
• $h_{12}$ = Reverse voltage gain [unitless]
• $h_{21}$ = Forward current gain [unitless]
• $h_{22}$ = Output admittance [Siemens]