expressions are equivalent if we define the complex amplitudes X as Xk = Akejk. 4.2.1 Write the Function M-file x(t) = Xke N x(t) = Ak cos(2πfkl+ok) k=1 (7) (8) Write an M-file called syn.sin.m that will synthesize a waveform in the form of (7). Although for loops are rather inefficient in MATLAB, you must write the function with one loop in this lab. The first few statements of the M-file are the comment lines-they should look like: function SYN_SIN usage: $ [xx, tt] 3 fk Xk 鲁 8 fs % dur tstart xx tt [xx, tt] = syn_sin (fk, Xk, fs, dur, tstart) Function to synthesize a sum of cosine waves syn sin (fk, Xk, fs, dur, tstart) vector of frequencies (these could be negative or positive) vector of complex amplitudes: Amp*e (j*phase) the number of samples per second for the time axis. total time duration of the signal starting time (default is zero, if you make this input optional) vector of sinusoidal values vector of times, for the time axis Note: fk and Xk must be the same length. Xk (1) corresponds to frequency fk (1), Xk (2) corresponds to frequency fk (2), etc. The MATLAB syntax length (fk) returns the number of elements in the vector fk, so we do not need a separate input argument for the number of frequencies. On the other hand, the programmer (that's you) should provide error checking to make sure that the lengths of fk and Xk are the same. See help error. Finally, notice that the input is defines the number of samples per second for the cosine generation; in other words, we are no longer constrained to using 20 samples per period. Include a copy of the MATLAB code with your lab report. 4.2.2 Default Inputs You can make the last input argument(s) take on default values if you use the nargin operator in MATLAB. For example, tstart can be made optional by including the following line of code: if nargin<5, tstart=0, end %--default value is zero 4.2.3 Testing In order to use this M-file to synthesize harmonic waveforms, you must choose the entries in the frequency vector to be integer multiples of some desired fundamental frequency. Try the following test and plot the result. [xx0, tt0] syn sin([0,100,250], [10,14*exp(-j*pi/3), 8*j1, 10000, 0.1, 0); *-Period = ? Measure the period of xx0 by hand. Then compare the period of xx0 to the periods of the three sinusoids that make up xx0, and write an explanation on the verification sheet of why the period of xx0 is longer.