Compute the Fourier series coecients (call them {a_k}) of the signal: x(t) = 6 c

Question

Compute the Fourier series coecients (call them {a_k}) of the signal:

x(t) = 6 cos (2pi*t + pi/4) + 4 cos((pi/6)*t -(pi/3)

What is the power contained in the signal x(t)?

Lety(t) = x(t -3)denote a time-shifted version of x(t), and let {b_k} denote the Fourier series coecientsof y(t). According to the time-shift property of the Fourier series, how should thecoecients {b_k} relate to the coecients {a_k}?

What is the power contained in y(t)?

Now let y(t) = 4x(t), according to the scaling property of the Fourier series, how shouldthe coecients {b_k} relate to the coecients {a_k}?

What is the power contained in this redefined y(t)?

Explanation / Answer

This is a heavy question as power is often desired by many. The power of money gives one options. And having power also gives one options. But the problem with having power is that it is so easily corrupted. There is a fine line between the proper use of power and the improper use of power. Corrupting power is quite tempting, particularly if you don't feel that you even have any power in your life. This is why we see "control freaks" and regular everyday people making life a little difficult for others. From the cashier that won't accept your coupon to the assistant manager making ridiculous schedules for a young worker with never a Friday or Saturday night to have fun with their friends. Such small things, but when power is in the hands over others, that temptation to abuse what power they have so easily just "takes over" for no more reason than just "because they can." I have real issue with one that abuses the power that is given to them. From policemen, lawyers, judges,and the worst, politicians, abuse and corrupt the power given to them by their station of life or by being voted into office. Once that power is available most find it too tempting NOT to cross that fine line into corruption. From pocketing a winning lotto ticket from its rightful owner to lining ones pocket with hard-earned taxpayers dollars instead of using the money as it should be used. To help our needy citizens and especially our children, our treasures in life. Our future. Slid into the pocket of a wealthy, corrupt politician instead of helping ones that need the resources to survive. Hungry children forsaken by those voted into office to feed and shelter our children. There are many karmic blunders but to take food out of the mouths of hungry children is a terrible horrific karmic debt that won't find redemption in this lifetime. There will be many politicians paying for their misdeeds for many a lifetime. Attempting in vain to learn that the corruption of power is a huge karmic error with severe retribution. Karma takes no prisoners. Karma is real and balances the wrongdoings of ones that abuse power unmercifully and cause the suffering of others. There is a steep price to pay for this "doing wrong" and "knowingly doing wrong." I have strong feelings regarding those that make new negative karma that hurt our precious children. I have much anger and possibly an (in)ability to forgive these horrible "people" that do such harm to others. Causing much suffering. Unnecessary suffering. So power is not "everything" to me as I do my level best to not manipulate another at my age of 55 years. It IS "one thing" because I am disturbed at how power is corrupted at each and every level of being. And as one that hasn't been given much in the way of power over others, I am choosing to agree with #3. Power is nothing.

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