Electrical Engineering 20N - Fall 2000 - Midterm 1

# Electrical Engineering 20N - Fall 2000 - Midterm 1 - 1...

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1 EECS20n, Solution to Midterm 1, 10/20/00 1. 10 points The function x : Reals Reals given by t Reals x ( t )=s in(2 π × 440 t ) is a mathematical example of a signal in the signal space [ Reals Reals ] . Give mathematical examples of signals in the following signal spaces. (a) (2 pts) [ Ints Reals ] n Ints ,x ( n )= n (b) (2 pts) [ Nats 0 EnglishWords ] n Nats 0 ( n the (c) (2 pts) [ Reals Reals 2 ] t Reals ( t )=( t, 2 t ) (d) (2 pts) [ { 0 , 1 , ··· , 600 }×{ 0 , 1 , , 400 }→{ 0 , 1 , , 255 } ] ( m,n ) ∈{ 0 , 1 , , 600 0 , 1 , , 400 } ( m + n ) mod 255 (e) (2 pts) Give an example of a practical space of signals whose mathematical representa- tion is [ { 0 , 1 , , 600 0 , 1 , , 400 0 , 1 , , 255 } ] . This is the appropriate signal space for images of size 600 × 400 pixels with an 8-bit color map index.

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2 2. 10 points The function H :[ Reals + Reals ] [ Nats 0 Reals ] given by: x [ Reals + Reals ] , n Nats 0 ,H ( x )( n )= x (10 n ) , is a mathematical example of a system with input signal space [ Reals + Reals ] and output signal space [ Nats 0 Reals ] . Give mathematical examples of systems whose (a) 5 pts input and output signal spaces both are [ Nats 0 Bin ] . The simplest example is the identity system: x, n, H ( x )( n x ( n ) . (b) 5 pts input signal space is [ Nats 0 Reals ] and output signal space is [ Nats 0 →{ 0 , 1 } ] . A simple example is a system which converts negative values to 0 and positive values to 1: x, n, H ( x )( n ( 0 , if x ( n ) 0 1 , if x ( n ) > 0 (c) 5 pts input signal space is [ Ints Reals ] and output signal space is [ Reals Reals ] .
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Electrical Engineering 20N - Fall 2000 - Midterm 1 - 1...

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