Assignment #09 - Solution Manual

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1 Assignment #09 - Solution Manual 1. Choose the correct statements about representation of a continuous signal using Haar wavelets. 1.5 points The signal is approximated using sin and cos functions. The signal can be approximated using step functions. Any continuous signal can be exactly represented using finite levels of Haar decomposition. Since the scaling function φt) is non-zero in t [0, 1], the signal can be represented using Haar wavelets only if it is non-zero in t [0, 1] and zero otherwise. The continuous signal is approximated using step functions as the scaling function φt) is a finite duration step function. Since step functions have discontinuities, continuous signals lie a ramp signal ft) t cannot be represented using finite levels of Haar decomposition. The time translates of scaling function can be used to represent the signal outside t [0, 1]. So, the signal can span over entire time duration t, ) for Haar representation.. True/False) If xt) is in V 0 and yt) is in V 1, then yt) xt) is in W 1. False 1.5 points We now that V 0 V 1, x V 0 and y V 1, yt) xt) V 1. Since V 1 W 1 V, the only common element between V 1 and W 1 is the 0 function and therefore, yt) xt) / W 1. Hence the given statement is False. 3. Which of the following equations are true with respect to the Haar wavelet decomposition? 1.5 points φ j 1 t) φ j t) φ j t 1) ψ j 1 t) φ j t) φ j t 1) φt 3) φt ) + φt 7) ψt ) φt 4) + φt 5) We have the friendly equations in the video lectures as given below: φ j 1 t) φ j t) + φ j t 1) 1) ψ j 1 t) φ j t) φ j t 1) ) For option 3, substituting j 1 and t t 3 in 1), we get φt 3) φt 3)) + φt 3) 1) φt ) + φt 7) 3) Similarly, for option 4, substituting j 1 and t t in ), we get ψt ) φt )) φt ) 1) φt 4) φt 5) 4) Therefore, only option and 3 is True. Page 1 of

2 4. The j th scale approximation of a signal f t) using Haar wavelets can be written in two forms 1.5 points as Choose the correct statements. ) a j) c j) + cj) +1 ) a j) c j) cj) +1 ) b j) c j) cj) +1 b j) c j) 1 cj) ) a j) φ j 1 t ) + c j) φ j t ) The answer is derived in the lecture. We now Therefore, + b j) ψ j 1 t ) φ j 1 t ) φ j t ) + φ j t + 1) ) and ψ j 1 t ) φ j t ) φ j t + 1) ) φ j t ) 1 [ φ j 1 t ) + ψ j 1 t )] 5) φ j t + 1) ) 1 [ φ j 1 t ) ψ j 1 t )]. ) c j) φ j t ) c j) φ j t ) + }{{} even terms c j) 1 c j) +1 c j) +1 φ j t + 1) ) } {{ } odd terms [ φ j 1 t ) + ψ j 1 t )] Using 5) and )) 1 [ φ j 1 t ) ψ j 1 t )] c j) + cj) +1 Comparing the above equation against φ j 1 t ) + a j) φ j 1 t ) + c j) cj) +1 b j) ψ j 1 t ), ψ j 1 t ). 7) we get a j) 1 ) c j) + cj) +1, b j) 1 ) c j) cj) +1. Page of

3 5. Given points 1, 0 t < 0.5 0, 0.5 t < 0.5 ft) 3, 0.5 t < 0.75, 0.75 t < 1 The Haar wavelet decomposition of the signal ft) is given by ft) φt) + ψt) + 1 ψt) + 5 ψt 1) ft) φt) + 1 ψt) + 3 ψt 1) ft) φt) + 1 ψt) + 5 ψt 1) ft) φt) + ψt) + 1 ψt) + 5 ψt 1) The smallest time is 1/4 time units and hence we start the decomposition with V. We now that V V 0 W 0 W 1. We need the coefficients in each of the above subspace. ft) φ4t) + 3φ4t ) φ4t 3) V Using the identities obtained in class, we have, φ4t) φ4t ) φ4t 3) φt) φt 1) Expanding the signal, we get φt) + ψt) φt 1) + ψt 1) φt 1) ψt 1) φt) + ψt) φt) ψt) ft) φt) + 1 }{{} ψt) + 5 ψt 1) }{{ } V 0 W 1. In question 5, what is the signal dimension of ft) in a space spanned by Haar scaling function 1.5 points φt), Haar wavelets ψ i t) and their time shifted version? 3 As ft) φt) + 1 ψt) + 5 ψt 1), the signal dimension is In question 5, let ˆft) be the signal obtained if the subspace corresponding to the details at 1 point the highest resolution is nulled out. What is ˆft)? ˆft) φt) ˆft) φt) + ψt) Page 3 of

4 ˆft) φt) + 1 ψt) ˆft) φt) + 1 ψt) Nulling out the signal in W 1, we have ˆft) φt). 8. From questions 5 and 7, percentage of the energy lost in representing the signal ft) as ˆft) points is 93 %. Round it to nearest integer). Energy in signal ft) after decomposition is given by E[ft)] ) ) E[ ˆft)] 1 dt ) ) ) ) 5 dt Percentage of energy lost energy is lost. E[ft)] E[ ˆft)] E[ft)] Almost 93% of the 9. We define a sequence of spaces as V Span { sin π t ), cos π t )}) for points, 1, 0, 1,. Choose the correct statements. The spaces V, Z satisfy the nesting property. V {0}. The spaces V, Z satisfy the scaling property. None of the above. V and V +1 are orthogonal spaces. Therefore, the spaces do not satisfy nesting property. Since V and V +1 are orthogonal spaces, V V Since 0 V Z, {0}. V If f t) V, we can write f t) a sin π t ) + b cos π t ) f t ) a sin πt) + b cos πt) V 0. Therefore, scaling property is satisfied. 10. Any function f t) L R) can be approximated using j th scale of Haar scaling function 1.5 points as coefficients a j)? a j) φ j t ). Which of the following expression is used to calculate the Page 4 of

5 a j) j +1) f t) dt j a j) ft),φj t ) φ j t ),φ j t ) a j) f t), φ j t ) a j) 1 j j +1) j f t) dt We now that the scaling functions are orthogonal and hence we consider the projections. Given a j) φ j t ), φ j t ) a j) φ j t ) f t), φ j t ) φ j t ), φ j t ) j +1) j a j) 1 j dt j j +1) j f t) dt 11. The signal f t) 3t + 4 is approximated using the j th scale approximation of Haar wavelets points given by a j) j + 1 ) + a j) 3 j + 1 ) + 4 a j) 3 j + 1) + a j) j + 1 4) 3 a j) φ j t ). Choose the correct statements. From question 10, we now j a j) j +1) j f t) dt [ ] 1 j +1) [ ] 1 j +1) 3t + 4) f t) 3 j j 1 f j + 1) ) f j ) ) 1 [ f j + 1) ) + f j )] [ f j + 1) ) f j )] Page 5 of

6 1 [ 3 j + 1) j + 4 ] [ 3 j + 1) j 4 ] j a j) 1 [ 3 j + 1) + 3 j + 8 ] [ 3 j] a j) 1 [ 3 j + 1) + 3 j + 8 ] 3 j + 1 ) Using question 11, what is the value of 8a ) 4 3a 4) 3? points From problem 11, a j) 3 j + 1 ) + 4 a ) ) + 4 a 4) ) + 4 8a ) 4 3a 4) 3 8 [ 3 4.5) + 4 ] 3 [ ) + 4 ] 90 Page of