**Problem 1**

a) 

Started from the correct definition of the inner product -> 1p

The signals/Fourier transforms manipulated using the correct definition of the (inverse) Fourier integral transform -> 1p

2 points from the correctness of the rest of the proof. The proof from the lecture slides is the easiest way to go, but one could also write both signals in the inner product out as Fourier transforms, combine the exponential functions and change the order of integration to find a Dirac delta distribution. The result then follows from the integral property of the delta distribution. Note that in this case, one must have different integration variables in the two Fourier integrals, e.g. t and u.

=> A total of 4 points from a)


b)

The conservation of energy in the Fourier integral transform follows from the previous result, since the energy of a signal is the inner product of the signal with itself.

-> 2p


**Problem 2**


a) Due to the mistake in the assignment (the signal provided was not 1-periodic, but 2-periodic), analyzing the signal as 1-periodic is acceptable.

The easiest way was to use the complex representations for sine and cosine and write the signal as a linear combination of complex exponential functions. The Fourier coefficients can then be read from the coefficients of the exponential functions at appropriate frequencies. Note that this results in some rational frequencies due to the wrong periodicity.


One could also calculate the coefficients straight from the definition by integrating over a single period of s(t). Note that here integrating over any interval of length 1 is acceptable (Actual period is 2 but again, spotting the 2-periodicity is not required). When integrating, care must be taken to not end up with 1/0-terms and such frequencies must be considered separately.


=> Total of 2 points from a)


b)

As the integration route in a) leads to an unreasonably demanding summation, it was sufficient to identify the correct definition for the energy computed from the Fourier coefficients. Deductions from wrong definitions (energy as an integral over the discrete Fourier coefficients etc.).

=> Total of 2 points from b)


c)

Similarly to b), the correct definition of DTFT suffices, as the coefficients gotten from integration lead to an untractable summation. Alternatively, one could use $$G(F(s))(t)=s(-t)$$, where F denotes the Fourier coefficient transform and  G is the discrete-time Fourier transform.

=> Total of 2 points from c)


**Problem 3**

For $$\hat{s}:$$
When the you have used the right formula => 1p
When all of the calculations have been calculated correctly => 1p
When the final result is correct => 1p

Same for $$\hat{\hat{s}}$$

For example if everything is calculated correctly but the final answers are incorrect (due to copying error for example) then you will be given 4p

**Problem 4**

For a)

When the initial formula used is used correctly => 1p
When all of the calculations and integration are done correctly => 1p
When the final is correct (either s(t)^2 or s(sqrt(2)*t)^2 or exp(-2*pi*t^2) ) => 1p


When the initial formula used is used correctly => 1p
When all of the calculations and integration are done correctly => 1p
When the final is correct (either 1/sqrt(2)*s(t/sqrt(2)) or 1/sqrt(2)*exp(-pi*t^2/2) or s**s(t) (** means convolution) )=> 1p

If all the calculations are done correctly, but you have done an incorrect variable change, I considered that only as the final answer being incorrect.


Last modified: Monday, 6 May 2019, 1:49 PM