Signals and Systems | Fourier Transform Does x(t)e -j2πft Reveal Frequency? Understanding the Fourier Transform The expression x(t)e -j2πft is an important part of the Fourier Transform. It is used to analyse the frequency components present in a continuous-time signal. However, this expression alone does not reveal the complete frequency spectrum. We must integrate it over time to obtain the Fourier Transform X(f). 1. What Is the Fourier Transform? The Fourier Transform is a mathematical operation that converts a signal from the time domain into the frequency domain. It helps us understand which frequencies are present in a signal and their corresponding magnitude and phase. Continuous-Time Fourier Transform Formula X(f) = ∫ −∞ ∞ x(t)e −j2πft dt Meaning of E...

  Q. Consider a real baseband signal: x(t) = e -2t ,   t ≥ 0 If 99% of the energy of x(t) lies within B Hz, which of the following options is TRUE for the value of B? (A)   B > 1 kHz (B) 63/π Hz < B < 64/π Hz Correct (C)   126/π Hz < B < 128/π Hz (D)   B < 1 Hz Step-by-Step Solution Step 1: Find the Fourier Transform X(f) = ∫₀∞ e -2t e -j2πft dt X(f) = ∫₀∞ e -(2+j2πf)t dt X(f) = 1 / (2 + j2πf) |X(f)|² = 1 / (4 + 4π²f²) Step 2: Calculate Total Energy E = ∫₀∞ |x(t)|² dt E = ∫₀∞ e -4t dt E = 1/4 ...

  Q. What is the 10′𝑠 complement of (47)10?   (A) 52   (B) 53   (C) 54   (D) 55 Solution 9's complement of 47 is equal to (99-47) = 52 So,  10's complement of 47 is equal to 52+1 = 53

Q. A QPSK modulated signal from an additive white Gaussian noise (AWGN) channel is received with an E b /N 0 = 8.4 dB at the input of a coherent QPSK demodulator. A maximum-likelihood reception method is used in the demodulator. Assume the complementary error function: erfc(u) ≈ 1 u√π exp(−u²) Which is the nearest bit error rate (BER) at the output of the demodulator? (A) 10 −3 (B) 10 −4 Correct (C) 10 −5 (D) 10 −6 Step-by-Step Solution Q(x) = 1/2*erfc(x/√2) Step 1: Convert SNR from dB to linear scale Given: E b /N 0 = 8.4 dB E b /N 0 = 10 8.4/10 = 10 0.84 = 6.918 ...

  Consider a real signal \(x(t)\), such that \(x(t)=\begin{cases} 0, & t Let the energy of the signal be defined as: \[ E[x(t)]=\int_{-\infty}^{\infty}|x(t)|^2dt \] Which of the following options correctly represents the ratio? \[ \frac{E[x(t)]}{E[3x(-3t+5)]} \] (A)   3 (B)   1 (C)   1/3 (D)   1/9 Step-by-Step Solution Step 1: Calculate the energy of x(t) The energy of a signal is given by: \[ E[x(t)]=\int_{-\infty}^{\infty}|x(t)|^2dt \] \[ =\int_0^1 2^2dt \] \[ =4[t]_0^1 \] \[ \boxed{E[x(t)]=4} \] Step 2: Calculate the energy of 3x(-3t+5) Using the energy t...

  Computer Organization | Number System | GATE Preparation How to Convert 262144 Decimal to 40000 Hexadecimal? Learn how to convert decimal numbers into hexadecimal numbers using the repeated division-by-16 method. This tutorial explains the conversion of 262144 decimal into 40000 hexadecimal with step-by-step calculations, a division table, verification and shortcut techniques for GATE Computer Science examinations. Number System Conversion Question: Convert Decimal to Hexadecimal Convert the decimal number 262144 into its equivalent hexadecimal number. Given Decimal Number 262144 10 1. Understand the Hexadecimal Number System The hexadeci...