1 (a)
Determine the fundamental period of the following signals:- \[ i) \ x(t) = \cos \dfrac {\pi}{3} t+ \sin \dfrac {\pi}{4} t \\ ii) \ x[n] = \cos ^ 2 \dfrac {\pi}{8}n \]
4 M
1 (b)
State and prove Time Shifting and Time Scaling property of continuous time Fourier Transform.
4 M
1 (c)
For the following system, determine whether it is. (i) memory less, (ii) causal, (iii) linear, (iv) time-invariant y[n]=x[n2]
4 M
1 (d)
Find out even and odd component of the following two signals:
\[ i) \ x(t) = t^3 +3t \\ ii) x[n] = \cos n+ \sin n + \cos (n) \sin (n) \]
\[ i) \ x(t) = t^3 +3t \\ ii) x[n] = \cos n+ \sin n + \cos (n) \sin (n) \]
4 M
1 (e)
Determine whether the signals are power of energy signals. Calculate energy/power accordingly:
i) x(t) = 0.9 e-3t u(t)
ii) x[n]=u[n]
i) x(t) = 0.9 e-3t u(t)
ii) x[n]=u[n]
4 M
2 (a)
Find the inverse Laplace Transform of \(\dfrac {s-2}{s(s+1)^3}\)
5 M
2 (b)
Let x(t)=1.......0 ≤ t≤ 2T and; h(t)=e-at.... 0≤ t≤ T. Compute y(t) using graphical convolution approach.
10 M
2 (c)
State and discuss the properties of the region of convergence for z-transform.
5 M
3 (a)
An LTI system is characterized by the system function: \[ h(z) = \dfrac {z} { \left ( z- \dfrac {1}{4} \right ) \left ( z+ \dfrac {1}{4} \right ) \left ( z- \dfrac {1}{2} \right )} \] write down possible ROCs. For different possible ROCs, determine causality and stability and impulse response of the system.
10 M
3 (b)
Calculate Z transform of the following signals: \[ i) \ x[n] =n \left ( - \dfrac {1}{4} \right )^n u[n]\times \left ( - \dfrac {1}{6} \right )^{-n} u [-n] \\ ii) \ x[n] = u[n-6] -u [n-10] \]
10 M
4 (a)
For the periodic signal x(t)=e-t with a fundamental period T0=1 second. Find the exponential form of Fourier Series. Also plot the Fourier spectrum (Magnitude and phase spectrum).
10 M
4 (b)
Consider a continuous time LTI system described by \[ \dfrac {dy(t)}{dt} + 2y(t)=x(t) \] Using the transform, find out output to each of the following input signals.
i) x(t)=e-t u(t)
ii) x(t)=u(t)
i) x(t)=e-t u(t)
ii) x(t)=u(t)
10 M
5 (a)
Convolute \[ x[n] = \left ( \dfrac {1}{3} \right )^n u[n] \ with \ h[n]= \left ( \dfrac {1}{2} \right )^n u[n] \] using convolute sum formula and verify your answer using z transform.
10 M
5 (b)
Explain Gibb's phenomenon. Also explain conditions necessary for the convergence of Fourier Series.
5 M
5 (c)
A system is described by the following difference equation. Find out its transfer function H(z). \[ y[n]= \dfrac {3}{4} y [n-1] - \dfrac {1}{8} y[n-2]+ x[n]+ \dfrac {1}{2} x [n-1] \]
5 M
6 (a)
For the signal x(t) depicted in the figure given below, sketch the signals:
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAT8AAAEFCAYAAACVR+BgAAAABHNCSVQICAgIfAhkiAAAAAlwSFlzAAALEwAACxMBAJqcGAAACOdJREFUeJzt3b+vpFUZB/Dv1V1WKaCBYKuxMtls6KSiAUw0dFJAYjbZxJLQLCyVt7DzP1jsIDHGhIKaGG2MfwGFVFupREKQZX8lLmMxMzDcvT/nzrxnznk+n+TN3ZnMMs95mfOd58yZu28CAAAAAAAAAAAAAADFXUzy7uInQBk3ktxN8nbrQgCmdC/JLMn91oVQ13daF0A5N5J8tfjzw+j+gCKWXd/y0P3RhM6PKR3s+pY/dX/A0O4n+SLJJ5l3fZ8sbj9oWRTANr2Q5LMkVxe3Z4ufVxf3v9iiKICpzU5+CGyPz/yAkoQfUJLwA0oSfkBJwg8oSfgBJQk/oCThB5Qk/ICShB9QkvADShJ+QEnCDyhJ+AElCT+gJOEHlCT8gJKEH1CS8ANKEn5AScIPKEn4ASUJP6Ak4QeUJPyAkoQfm/RskveSfJjk9SRPty0Hjib82JQnMw+9fyV5K8nlJH9sWhHABH6W5G8rt3+Q5KskTx3x+NnWKwKYwBNJnlm5/UqSj455vPADhvNaks+SvHrMY4QfMIxLSd5P8nGS54574P7+/mr4zVaObPn+/QO3z3v85bhxAuO7kOSDJDczD8FjzWazUTq/UcYBrOnnmX/G90SSx1eOvSMeP0pojDIOYE2/zeHLwh8d8fhRQmOUcQATGSU0RhlHOb7kTBPjfORHr4QfUJLwA4AJjbLuHWUc5ej8gJKEH1CS8KMJu720JvyAkoQfAExolHXvKOMoR+cHlCT8gJKEH03Y7aU14QeUJPwAYEKjrHtHGUc5Oj+gJOEHlCT8aMJuL60JP6Ak4QcAExpl3TvKOMrR+QElCT+gJOFHE3Z7aU34ASUJPwCY0Cjr3lHGUY7ODyhJ+AElCT+asNtLa8IPKEn4AcCERln3jjKOcnR+QEnCDyhJ+NGE3V5aE35AScIPACY0yrp3lHGUo/Nj191McqV1EQCbclLH9HKSdxaPe3775axN59cpnR9NnGK393KSO0lub78agInMTv9dl1vR+bEFOj+gJOHXn/082m3MVo4u7t/b28sg/tq6AKhilGXWKMveZJz/J6Xo/OD89iIAuyP8YDMEIHAqowbFqOMCNmTkkBh5bMA5jR4Qo48PJjfKpBplHMepMMZu2fCA7bEJssOEH2yXAAS+pVogVBsvcISKYVBxzMABVYOg6rjh3EaZPKOMYx2Vx74zbHjA9GyC7ADhB20IQCjKxJ9zHqAYk/4bzgUUYsJ/m/MBJxhlkowyjk1yTiZkwwN2h02QCQk/2C0CEAZngh/P+YFBmdwnc45gQCb26ThPsDDKZBhlHFNwrrbAhgfsPpsgWyD8oA8CEAZhIq/HeYPOmcTrc+6gYybw+Th/lDPKi36UcbTkHJ6DDQ/ol02QcxB+0DcBCJ0xYTfL+eRIzyZ5L8mHSV5P8nTbciZzM8mV1kUcwmTdPOeURzyZ5NMkv8s8BN9J8uemFW3fy5mPc5bk+ca1HMZE3Q7n9ZQutC5gIj9N8o8kby1u/ybJP5M8lXko9mSW+ec8J7mc5E6S29sthx2z/AzwNK8RCngiyTMrt19J8lGjWs7rrO/st6Lzq8j55RGvJfksyautC1mT8OO0nOOC9pM8XBxvLO67lOT9JB8nea5RXdty2HiXbkX4VeY8F3chyQeZ73xealzL1G5F+FXnXB+iyobHS0l+nORXSb6b5PHF/ffihcH4bIIcospveDyX5CdJ/pv5Dujy+GHLotYkrFmH3wShe6O8gEcZx9J+6wJOabTzTiGjvHhHGcdST+Ppqdad9kLrAujSaBOwt/H0Vu/Si60LSJIbSe5n/p05OKteJ99RehxPjzV/nvlG5fUWT35j8eR3knyR5GqLIuhejxPvOL2Op7e6r2X+K5t3ktzNBCF4Md8Ovdni+GTbT8wjenuxHmWUcSz1PJ7eav9Pvsmg1RC8eJb/yGm/9/Nukl8m+f7KfQ8z/84cwNQO5s/dJH9I8utNP9HFJG9n/hlfz51fb+9whxlhDMk441gabTy7bLXz+zLzFembSR7b9hOvhqDP/FjXaGEx2nh21fIzv9XQm9zbSR7Ebi/rGS0sRhvPrvo88+arSegdtBPfu6E7o4VFL7/h0buXWhcwih4nYI81H2aUcSzt6rVSTlL12jakv0nYW71HGWUcu36tlONUvLbN16r8k1bH8c/9cB49XytlpGvbQDdG6fyWbqW/zm+ka9tAN4Tfbun92jZnZtkLtV3K/DcjLif5RZK/ty1nOsLvcLv8GeAu10ZfLiT5U5J/Zx5+D9qWMy3hdzibIFRQ+to2Va7hsQ7XPGB0I13bBrrhjYWmdH5AScIPKEn4nc0uLNV2oQbonvA7G5sgMAjhd3YCEIC1eQOhKZ0fUJLwA0oSfpsx5RLOchE2QPhthk0Q6Izw2xwBCMCJvFHQlM4PKEn4ASUJv+3axtLOchE2QPhtl00Q2FHCb/sEIABf84ZAUzo/oCThB5Qk/No4z5LPchE2QPi1YRMEGhN+7QhAgIIEP03p/ICShB9QkvDbLadZClouwgYIv91iEwQmIvx2jwAEGJiApymdH725meRK6yIA1nXWzu/lJO8s/t7zmy+HanR+fZgd8edKLie5k+R260KAac0O/OzduuO4FZ0fG6Dz64ddYNgg4deXvcXRo/3Mw3t55MDt/QOPfbg43piwRoCts+ylKZ0fUJLwA4AJ2byhKZ0fUJLwA0oSfkBJwg8oSfgBJQk/oCThB5Qk/ICShB9QkvADShJ+QEnCDyhJ+AElCT+gJOEHlCT8gJKEH1CS8ANKEn5AScIPKEn4ASUJP6Ak4QeUJPyAkoQfUJLwA0oSfkBJwg8oSfgBJQk/oCThB5Qk/ICShB9QkvADShJ+QEnCDyhJ+AElCT+m8mKSz5NcO3D/tcX9L01eEaXttS6AUu4l+V+S+0meSvJpku8lubj4CTCk60nuJJllHoKzJF8mebNlUQBTuJt56C2Pe23LAZjGaven6wNKWXZ/uj6glOuZB6CuDyjlYpLfJ3msdSEAAAAAAAAAAAAAAAAAAAAAsHH/B4XP9zwwZRdjAAAAAElFTkSuQmCC)
i) x(-t)
ii) x(t+6)
iii) x(3t)
x(t/2)
i) x(-t)
ii) x(t+6)
iii) x(3t)
x(t/2)
10 M
6 (b)
For the periodic signal x[n] given below, find out Fourier Series coefficient: \[ x[n]= 1+ \sin \left ( \dfrac {2 \pi } { N} \right )n + 3 \cos \left [ \dfrac {2 \pi}{N} \right ]n+ \cos \left ( \dfrac {4\pi}{ N} n + \dfrac{\pi}{2} \right ) \]
10 M
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