Step 1: The twiddle factor in the FFT is defined as: \[ W_N = e^{-j\frac{2\pi}{N}} \] where \( W_N^k \) represents the \( k \)th power of the twiddle factor for an \( N \)-point FFT.
Step 2: Given: \[ W_4^1 = W_x^2 \] Substituting the definition of the twiddle factor: \[ e^{-j\frac{2\pi}{4} \times 1} = e^{-j\frac{2\pi}{x} \times 2} \]
Step 3: Simplifying: \[ e^{-j\frac{2\pi}{4}} = e^{-j\frac{4\pi}{x}} \] Equating exponents: \[ \frac{2\pi}{4} = \frac{4\pi}{x} \] \[ \frac{\pi}{2} = \frac{4\pi}{x} \] Solving for \( x \): \[ x = 4 \]
Step 4: Evaluating options:
- (A) Incorrect: \( x = 2 \) does not satisfy the equation.
- (B) Correct: \( x = 4 \) is the correct answer.
- (C) Incorrect: \( x = 8 \) does not match.
- (D) Incorrect: \( x = 16 \) is incorrect.
Match List-I with List-II:
List-I (Modulation Schemes) | List-II (Wave Expressions) |
---|---|
(A) Amplitude Modulation | (I) \( x(t) = A\cos(\omega_c t + k m(t)) \) |
(B) Phase Modulation | (II) \( x(t) = A\cos(\omega_c t + k \int m(t)dt) \) |
(C) Frequency Modulation | (III) \( x(t) = A + m(t)\cos(\omega_c t) \) |
(D) DSB-SC Modulation | (IV) \( x(t) = m(t)\cos(\omega_c t) \) |
Choose the correct answer:
The bulking of the sand is increased in volume from 20% to 40% of various sand and moisture content ranges from ……… to ……….. percent.