# hw2 - ECE 562 Fall 2011 September 8 2011 HOMEWORK...

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Unformatted text preview: ECE 562 Fall 2011 September 8, 2011 HOMEWORK ASSIGNMENT 2 Reading: Madhow, Sections 3.3, 2.5, 2.6 Due Date: September 20, 2010 (in class) 1. Consider the following three waveforms: − 1 2 t 1 2 t 1 2 f 2 ( t ) f 3 ( t ) t f 1 ( t ) 1 2 4 4 2 4 3 1 − 1 2 (a) Show that these waveforms are orthonormal. (b) Express the following waveform s ( t ) as a linear combination of f n ( t ) , n = 1 , 2 , 3. − 1 4 1 2 3 1 − 1 2 s ( t ) 4 t 2 1 4 2. Gram-Schmidt. Determine an orthonormal basis for the span of the following four signals: c circlecopyrt V.V. Veeravalli, 2011 1 2 t 3 1 − 2 s 3 ( t ) t 2 s 1 ( t ) 3 t s 4 ( t ) t s 2 ( t ) 1 2 2 3. Signal Spaces as Vector Spaces. Let { f ℓ ( t ) } n ℓ =1 be an orthonormal basis for S (a) Suppose s ( t ) ∈ S has the representation in terms of this basis as: s ( t ) = n summationdisplay ℓ =1 s ℓ f ℓ ( t ) and define the column vector s = [ s 1 s 2 ··· s n ] ⊤ . Show that bardbl s bardbl = n summationdisplay ℓ =1 | s ℓ | 2 = radicalbig...
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hw2 - ECE 562 Fall 2011 September 8 2011 HOMEWORK...

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