Digital Signal Processing

The Representation of Bandpass Signals

Question 1
Marks : +2 | -2
Pass Ratio : 100%
What is the equivalent time domain relation of xl(t) i.e., lowpass signal?
\\(x_l (t)=[x(t)+j ẋ(t)]e^{-j2πF_c t}\\)
x(t)+j ẋ(t) = \\(x_l (t) e^{j2πF_c t}\\)
\\(x_l (t)=[x(t)+j ẋ(t)]e^{-j2πF_c t}\\) & x(t)+j ẋ(t) = \\(x_l (t) e^{j2πF_c t}\\)
None of the mentioned
Explanation:
\\(x_l (t)=x_+ (t) e^{-j2Ï€F_c t}\\)
Question 2
Marks : +2 | -2
Pass Ratio : 100%
If the signal ẋ(t) can be viewed as the output of the filter with impulse response h(t) = 1/πt, -∞ < t < ∞ when excited by the input signal x(t) then such a filter is called as __________
Analytic transformer
Hilbert transformer
Both Analytic & Hilbert transformer
None of the mentioned
Explanation:
The signal ẋ(t) can be viewed as the output of the filter with impulse response h(t) = 1/πt,
Question 3
Marks : +2 | -2
Pass Ratio : 100%
What is the equivalent time –domain expression of X+(F)=2V(F)X(F)?
F(+1)[2V(F)]*F(+1)[X(F)]
F(-1)[4V(F)]*F(-1)[X(F)]
F(-1)[V(F)]*F(-1)[X(F)]
F(-1)[2V(F)]*F(-1)[X(F)]
Explanation:
Given Expression, X+(F)=2V(F)X(F).It can be calculated as follows
Question 4
Marks : +2 | -2
Pass Ratio : 100%
Which of the following is the right way of representation of equation that contains only the positive frequencies in a given x(t) signal?
X+(F)=4V(F)X(F)
X+(F)=V(F)X(F)
X+(F)=2V(F)X(F)
X+(F)=8V(F)X(F)
Explanation:
In a real valued signal x(t), has a frequency content concentrated in a narrow band of frequencies in the vicinity of a frequency Fc. Such a signal which has only positive frequencies can be expressed as X+(F)=2V(F)X(F)
Question 5
Marks : +2 | -2
Pass Ratio : 100%
In time-domain expression, \\(x_+ (t)=F^{-1} [2V(F)]*F^{-1} [X(F)]\\). The signal x+(t) is known as
Systematic signal
Analytic signal
Pre-envelope of x(t)
Both Analytic signal & Pre-envelope of x(t)
Explanation:
From the given expression, \\(x_+ (t)=F^{-1} [2V(F)] * F^{-1}[X(F)]\\).
Question 6
Marks : +2 | -2
Pass Ratio : 100%
In equation \\(x_+ (t)=F^{-1} [2V(F)]*F^{-1} [X(F)]\\), if \\(F^{-1} [2V(F)]=δ(t)+j/πt\\) and \\(F^{-1} [X(F)]\\) = x(t). Then the value of ẋ(t) is?
\\(\\frac{1}{π} \\int_{-\\infty}^\\infty \\frac{x(t)}{t+τ} dτ\\)
\\(\\frac{1}{π} \\int_{-\\infty}^\\infty \\frac{x(t)}{t-τ} dτ\\)
\\(\\frac{1}{π} \\int_{-\\infty}^\\infty \\frac{2x(t)}{t-τ} dτ\\)
\\(\\frac{1}{π} \\int_{-\\infty}^\\infty \\frac{4x(t)}{t-τ} dτ\\)
Explanation:
\\(x_+ (t)=[δ(t)+j/πt]*x(t)\\)
Question 7
Marks : +2 | -2
Pass Ratio : 100%
If we substitute the equation \\(x_l (t)= u_c (t)+j u_s (t)\\) in equation x (t) + j ẋ (t) = xl(t) ej2πFct and equate real and imaginary parts on side, then what are the relations that we obtain?
x(t)=\\(u_c (t) \\,cos⁡2π \\,F_c \\,t+u_s (t) \\,sin⁡2π \\,F_c \\,t\\); ẋ(t)=\\(u_s (t) \\,cos⁡2π \\,F_c \\,t-u_c \\,(t) \\,sin⁡2π \\,F_c \\,t\\)
x(t)=\\(u_c (t) \\,cos⁡2π \\,F_c \\,t-u_s (t) \\,sin⁡2π \\,F_c \\,t\\); ẋ(t)=\\(u_s (t) \\,cos⁡2π \\,F_c t+u_c (t) \\,sin⁡2π \\,F_c \\,t\\)
x(t)=\\(u_c (t) \\,cos⁡2π \\,F_c t+u_s (t) \\,sin⁡2π \\,F_c \\,t\\); ẋ(t)=\\(u_s (t) \\,cos⁡2π \\,F_c t+u_c (t) \\,sin⁡2π \\,F_c \\,t\\)
x(t)=\\(u_c (t) \\,cos⁡2π \\,F_c \\,t-u_s (t) \\,sin⁡2π \\,F_c \\,t\\); ẋ(t)=\\(u_s (t) \\,cos⁡2π \\,F_c \\,t-u_c (t) \\,sin⁡2π \\,F_c \\,t\\)
Explanation:
If we substitute the given equation in other, then we get the required result
Question 8
Marks : +2 | -2
Pass Ratio : 100%
What is the frequency response of a Hilbert transform H(F)=?
\\(\\begin{cases}&-j (F>0) \\\\ & 0 (F=0)\\\\ & j (F<0)\\end{cases}\\)
\\(\\left\\{\\begin{matrix}-j & (F<0)\\\\0 & (F=0) \\\\j & (F>0)\\end{matrix}\\right. \\)
\\(\\left\\{\\begin{matrix}-j & (F>0)\\\\0 &(F=0) \\\\j & (F<0)\\end{matrix}\\right. \\)
\\(\\left\\{\\begin{matrix}j&(F>0)\\\\0 & (F=0)\\\\j & (F<0)\\end{matrix}\\right. \\)
Explanation:
H(F) =\\(\\int_{-∞}^∞ h(t)e^{-j2πFt} dt\\)
Question 9
Marks : +2 | -2
Pass Ratio : 100%
In the relation, x(t) = \\(u_c (t) cos⁡2π \\,F_c \\,t-u_s (t) sin⁡2π \\,F_c \\,t\\) the low frequency components uc and us are called _____________ of the bandpass signal x(t).
Quadratic components
Quadrature components
Triplet components
None of the mentioned
Explanation:
The low frequency signal components uc(t) and us(t) can be viewed as amplitude modulations impressed on the carrier components cos2Ï€Fct and sin2Ï€Fct, respectively. Since these carrier components are in phase quadrature, uc(t) and us(t) are called the Quadrature components of the bandpass signal x (t).
Question 10
Marks : +2 | -2
Pass Ratio : 100%
What is the equivalent lowpass representation obtained by performing a frequency translation of X+(F) to Xl(F)= ?
X+(F+Fc)
X+(F-Fc)
X+(F*Fc)
X+(Fc-F)
Explanation:
The analytic signal x+(t) is a bandpass signal. We obtain an equivalent lowpass representation by performing a frequency translation of X+(F).