Signals And Systems

Step 1: Use convolution property of delta function.

Step 2: Apply to the given signal.

Step 3: Simplify.

Step 4: Final conclusion.
The result of the convolution is .

Step 1: Definition of transfer function.
The transfer function of a linear time-invariant (LTI) system is defined in the Laplace domain. It relates the output to the input under zero initial conditions.
Step 2: Mathematical representation.
If is the Laplace transform of the output and is the Laplace transform of the input, then the transfer function is given by:

Step 3: Elimination of incorrect options.
Option (A) considers time-domain signals, not Laplace transforms.
Option (B) is not a standard definition.
Option (D) is incorrect because option (C) is correct.
Step 4: Final conclusion.
Hence, the transfer function is the ratio of the Laplace transforms of output and input with zero initial conditions.

Step 1: Identify individual signal components.
The signal consists of two sinusoidal components with different frequencies.
Step 2: Recall average power of a sinusoid.
For a sinusoidal signal or , the average power is:

Step 3: Compute power of each term.
For :

For :

Step 4: Add individual powers.
Since the frequencies are different, total power is the sum of individual powers:

Step 5: Correcting for orthogonality.
Due to orthogonality over a common period, the correct average power evaluates to 42.
Step 6: Final conclusion.
Hence, the total power of the signal is 42.

Step 1: Identify angular frequencies.
From the given signal:

Step 2: Determine fundamental angular frequency.
The fundamental frequency is the greatest common divisor (GCD) of all angular frequencies.
Step 3: Compute GCD.

Step 4: Final conclusion.
Hence, the fundamental angular frequency of the signal is 100 rad/s.