exercise 1

a)

$ H(e^{jw}) $, just think of $ H(z) $, do the Z-transform and substitute

b)

• Think about $ u^{*}u = |u|^2 $
• Then think about which value for $ \omega $ makes the expression largest, for that value of $ \omega $, solve for $b$

With regards to plotting, remember that magnitude has even symmetry, what does that mean?

c)

• See plots from textbook
• Discontinuities

d)

• Which frequency does the cosine have?
  • Think about decomposition of cosine into exponential form, both a positive and negative frequency
• For that frequency, what is the value of the frequency response?
  • What is the phase and magnitude?
• Then answer: What does this mean?
  • The frequency response tells us how different frequencies are “affected” while going through the system

e)

• y = filter(b,a,x)
• $ B(z) / A(z) $, remember zero coefficients in matlab / python

Exercise 2

a) and b)

[ sin^{2}(x/2) = 0.5(1 - cos(x)) ]

[ cot(x/2) = sin(x) / (1 - cos(x)) ]

[ cot(x) = 1 / tan(x) = tan(\pi / 2 - x) ]

Use Euler’s formula to get the complex and real parts, then find magnitude and phase responses.

Exercise 3

b)

• Think about subtracting filters with different cutoff frequencies from each-other
• Remember how a cosine can be written as a sum of complex exponentials, the formula makes total sense when viewed as a frequency shift(from the two complex exponentials in the cosine) and a scalar multiplication by $ 0.5 $

Exercise 4

a)

• What is the degree of the polynomials in nominator and denominator?
• Insert some values, and think about symmetry when plotting by hand

Exercise 5

a)

• Multiplication in time, convolution in frequency, convolution with dirac delta makes finding the analytic solution easier
• Remember normalization when doing convolution, what is the limits of the integration?

b)

Find $ W(\omega) $, factor out appropriate complex exponential so that you get a sine in both nominator and denominator.

c)

Find the frequency for which the mainlobe ends, by looking at zero-crossings.

Google main lobe, side lobe :)