This question is in the form of an exercise and questions designed to give you more insight into signal processing. On the Moodle site for the module there is an EXCEL file called "FFT_exercise_2012". The first column contains the numbers 1 to 32. Then there is a column representing a "random time series", i.e. a digitised signal. The next eight columns are sine waves and cosine waves in pairs. The first pair is the fundamental based on 32 samples, i.e. one complete cycle. Then there are the second harmonics, and the third harmonics. The graphs for each of the digitised functions have been created to help you visualise these standard functions.

The time series given to you is the exact sum of a mean value plus a different multiple of each of these 8 sine and cosine components plus one further mystery harmonic. Your main task is to find these 10 values. But there are some initial tasks to help you think through how to find the values.

(a) Generate the columns on the EXCEL spreadsheet you will need for sin1, cos1, sin2, cos2, ... , sin4, cos4. Please note that the EXCEL function SIN(value) requires the value to be in radians. Also the EXCEL function PI() will give ??to lots of significant figures.

(b) Obtain the sum of the squares of values in each of the sine and cosine columns using EXCEL. What values do you obtain?

(c) What is the mean value of the random time series?

(d) What are the amplitudes of the 8 sinusoidal components that added to the mean make up the time series? Produce a table corresponding to sin1, cos1, sin2, cos2, etc.

(e) What is the amplitude of the mystery harmonic component and which component is it? Use the notation sin or cos plus a harmonic number, e.g. sin5, cos6, etc.