Fourier Transform of \(x(t)=\dfrac{t}{(1+t^2)^2}\) We want to determine the Fourier transform of $$ x(t)=\frac{t}{(1+t^2)^2} $$ using the Fourier transform definition $$ X(\omega)=\int_{-\infty}^{\infty}x(t)e^{-j\omega t}\,dt $$ Step 1 : Let $$ x_1(t)=\frac{1}{1+t^2} $$ Then $$ X_1(\omega)=\pi e^{-|\omega|}. $$ Proof: $$ \int_{-\infty}^{\infty}\frac{1}{1+t^2}e^{-j\omega t}\,dt $$ Use the Residue Theorem. $$ \int_{-\infty}^{\infty}\frac{1}{1+z^2}e^{-j\omega z}\,dz $$ Or, $$ \int_{-\infty}^{\infty}\frac{1}{(z+j)(z-j)}e^{-j\omega z}\,dz $$ Poles are at \(z=-j\) and \(z=j\). For the pole at \(z=-j\): $$ \operatorname*{Res}_{z=-j}\left(\frac{e^{-j\omega z}}{(z+j)(z-j)}\right) = \lim_{z\to -j}(z+j)\frac{e^{-j\omega z}}{(z+j)(z-j)} = \frac{e^{-\omega}}{-2j}. $$ Therefore, $$ -2\pi j\left(\frac{e^{-\omega}}{-2j}\right) = \pi e^{-\omega}. $$ For the pole at \(z=j\): $$ \operatorname*{Res}_{z=j}\left(\frac{e^{-j\omega z}}{(z+j)(z-j)}\righ...

  Effect of Noise in Amplitude Modulation (AM)  Understanding the effect of noise in amplitude modulation (AM) is one of the most important topics in Analog Communication Systems.  This article explains the complete derivation step by step, including: Transmitted AM signal Received signal equation Why noise is written in envelope-phase form Difference between r(t) and y(t) Envelope detector output Threshold effect Step 1: Transmitted AM Signal The transmitted amplitude modulated (AM) signal is s(t)=A c [1+k a m(t)]cos(2πf c t) where A c = Carrier amplitude m(t) = Message signal k a = Amplitude sensitivity f c = Carrier frequency Step 2: Noise is Added by the Channel Every communication channel introduces random noise. Let the channel noise be n(t) The received signal becomes r(t)=s(t)+n(t) This is the most fundamental equation in communication systems. Received Signal = Transmitted...

BER vs SNR: Constellation Analysis BER Derivation from Constellation Points The probability of a bit error in an AWGN channel is determined by the minimum Euclidean distance (\(d_{min}\)) between symbols in a constellation. The generic formula for Bit Error Rate (BER) is: \[ BER = Q \left( \frac{d_{min}}{\sqrt{2N_0}} \right) \] 1. Binary PSK (BPSK) BPSK is antipodal . The points are located at \(+\sqrt{E_b}\) and \(-\sqrt{E_b}\). Because the carrier is always "on" at full strength, the Peak Power equals the Average Power . Distance (\(d_{min}\)): \(2\sqrt{E_b}\) BER Formula: \(Q(\sqrt{2E_b/N_0})\) At 0 dB SNR: \(Q(\sqrt{2}) \approx 0.078\) 2. Orthogonal FSK FSK symbols are perpendicular in signal space. Like PSK, FSK has a constant envelope , meaning the power never fluctuates regardless of which frequency is sent. Distance (\(d_{min}\)): \(\sqrt{2E_b}\) BER Formula: \(Q(\sqrt{E_b/N_0...

Alamouti Scheme Alamouti's Space-Time Block Coding (STBC) is a fundamental technique in MIMO wireless communication used to achieve transmit diversity. In a 2x1 system, it allows the receiver to recover symbols effectively even under severe fading conditions without requiring channel state information at the transmitter. 1. The Precoding Matrix The Alamouti precoding matrix defines the symbols transmitted from two antennas over two consecutive time slots: X = [ s1 , -s2* ; s2 , s1* ] (Columns represent Time Slots, Rows represent Antennas) 2. Channel Model & Matrix Equations In a 2x1 MIMO system, let h1 and h2 be the complex channel gains from Transmit Antenna 1 and 2 to the receiver, respectively. The received signals y1 and y2 over two time slots are: ...

FSK Modulation Simulator Frequency Shift Keying (FSK) Modulation Carrier 1 Frequency (Hz): Carrier 2 Frequency (Hz): Sampling Rate (Hz): Bit Rate (bps): Execute FSK Demodulation Execute FSK Demodulation Simulation Mathematics Digital FSK Signal Equation: s(t) = A_c cos(2π f_i t) f_i = f_1 for bit '1', f_i...

Instructions for Frequency Shift Keying Modulation (FSK) Step 1: Click on 'Generate Message' button to generate input message signal Step 2: Then click on 'Generate Carrier' button to generate carrier signal Step 3: You can change the carrier signal frequencies from the input fields Step 4: Click on 'Simulate FSK' button to generate Frequency Shift Keying Signal Carrier Frequency 1 in Hz: Carrier Frequency 2 in Hz: ...

  Wide-Sense Stationarity (WSS) A process {X t } is Wide-Sense Stationary (or covariance stationary) if it possesses a finite second moment (E[X t 2 ] Constant Mean: E[X t ] = μ for all t . Time-Invariant Covariance: The autocovariance between any two observations depends solely on the lag τ: Cov(X t , X t+τ ) = γ(τ). (This implies a constant variance, where Var(X t ) = γ(0)). White Noise (WN) A process {ε t } is defined as white noise if it is a WSS process with E[ε t ] = 0, constant variance σ 2 , and Cov(ε t , ε t+τ ) = 0 for all τ ≠ 0. If the variables are also independent and identically distributed (I.I.D.), it is termed "Independent White Noise." LTI Systems and Discrete Convolution A stationary time series is characterized as the output of a Linear Time-Invariant (LTI) system driven by...

Output Statistics of an LTI System for WSS Input | Signal Processing Guide Statistical Properties of LTI System Output for Wide-Sense Stationary (WSS) Inputs In signal processing and communication theory, understanding how a Linear Time-Invariant (LTI) system transforms the statistical characteristics of a random process is fundamental. This guide explores the output response when the input is a Wide-Sense Stationary (WSS) process. System Definition: Suppose an input signal x(t) is a Wide-Sense Stationary (WSS) random process applied to an LTI system with an impulse response h(t) . The output is defined by the convolution integral: y(t) = x(t) * h(t) = ∫ x(t - τ) h(τ) dτ 1. LTI System Output Statistics Output Mean (Expected Value) The mean of the output process is a constant scaled by the system's DC gain. μ y = E[y(t)] = μ x H(0) Where H(0) = ∫ h(t) dt represents the frequency response at zero frequenc...