Homework 2
Section 6.5, 6.4(a): For f(t) = cos(ω0t)and ψ(t)a wavelet symmetric
about 0, verify that W f (u, s) = √sˆ
ψ(sω0) cos(ω0u).
Since ψis symmetric about 0, we have ψ∗=ψ. Now this verification is a
series of equalities in which we manipulate the variables to obtain the desired
statement with the rationale in brackets preceeding the step:
W f (u, s) = hf, ψu,si=1
√sZ∞
−∞
cos(ω0t)ψ∗t−u
sdt
[def of cos] = 1
2√sZ∞
−∞ eiω0t+e−iω0tψt−u
sdt
[y=t−u
s] = √s
2Z∞
−∞ eiω0syeiω0u+e−iω0sy e−iω0uψ(y)dy
=√s
2Z∞
−∞
ei(sω0)yeiω0uψ(y)dy +√s
2Z∞
−∞
e−i(sω0)ye−iω0uψ(y)dy
=√s
2eiω0uZ∞
−∞
ei(sω0)yψ(y)dy +√s
2e−iω0uZ∞
−∞
e−i(sω0)yψ(y)dy
[symmetry of ψ] = √s
2eiω0u+e−iω0uZ∞
−∞
e−i(sω0)yψ(y)dy
=√seiω0u+e−iω0u
2ˆ
ψ(sω0)
=√sˆ
ψ(sω0) cos(ω0u).
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