SOME QUESTIONS ON TRANSFORMS
AJ, 2009, PAB 2010, JW 2013
(1) Show that every signal x(n) = A cos (ωn + ϕ) can be written as a signal with
frequency −π < ω < π, and that it is periodic if and only if f = ωπ is rational. (Pp
15-16).
(2) Derive the expression (9.5) for Continuous time Fourier transform (CTFT) of a
sampled signal. Define bandwidth, formulate the sampling theorem and explain
why it is true. (P 396ff or lecture notes).
(3) With what frequency/angular velocity must a signal x(t) = A cos (ωt + ϕ) be sampled in order to avoid aliasing? Motivate your answer (see chapter 9.1, in parti p
396).
(4) Derive the result that every LTI-system y = S{x} can be written as the convolution
(sw. ‘faltning’) y(t) = x(t) ∗ y(t) (or y[n] = x[n] ∗ y[n] for DLTI systems) where h
is the impulse response. (See chap 10.4, in partiP
page 430-31)
∞
(5) Show that a DTLTI-system is BIBO-stable if
k=−∞ |h[k]| < ∞ and casual if
h[n] = 0, ∀n < 0. (Chapt 10.8 p 452-53)
(6) Derive the inversion formula for the Fourierseries of a periodic signal x(t). (Thm
11.2 p 467-8)
(7) Assume x[n] is a real signal. Show that X ∗ (ω) = X(−ω) and use that to show that
if x is even (odd) then the transform is purely real (purely imaginary).
DTFT
(8) Derive the formula for x[n−k] ←→ X(Ω)e−jΩk (for constant k). Derive the formula
DTFT
x[−n] ←→ X(−Ω)
DTFT
(9) Derive the formula x[n] ∗ y[n] ←→ X(Ω)Y (Ω).
n
DTFT dX(Ω)
(10) Derive the formula x[n] ←→
j
dΩ
(11) Derive Parseval’s formula. (Ch 11.5.13).
(12) Show how the discrete Fourier-transform can be derived from the discretisation
of the continuous Fourier transform. How is zero-padding applied in this case?
(Chapter 12.2 and 12.3.1)
(13) Let Ψ be a function with support in the interval [0, 1]. Memorise the formula
for the baby-Wavelets 2n/2 Ψ(2n t − k) (or derive it from scaling) and show that
kΨn,k k2 = kΨk2 and that the support is in the interval [(k − 1)2−N , k2−N ]. (see )
(14) Describe the Haar–wavelets on the interval [0, 1] and show that they form and
orthogonal set. Describe the wavelet-expansion for a function f ∈ L2 [0, 1]. (see )
(15) Define all the main concepts involved in multi-resolution analysis: Vector-space,
inner product space, Hilbertspace, and orthogonal subspace. (see )
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AJ, 2009, PAB 2010, JW 2013
(16) Define the multi-resolution analysis and the wavelet generated by MRA. For the
Haar-wavelet, what is V0 √
and W0 ? (see )
(17) Show that the functions 2ϕ(2t − k) form a basis for V1 . (see )
(18) Derive the scaling equation for ϕ and the corresponding equation for the wavelet.
What is the coefficients for the the Haar wavelet?
(19) Explain why the wavelets series gives localisation in time and frequency.