v The set of all distinct values that a discrete time sinusoidal sequence can

V the set of all distinct values that a discrete time

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(v) The set of all distinct values that a discrete time sinusoidal sequence can have occurs for values of ranging from or simply ] , [ Because of these last two properties, any sinusoid having the frequency is called an alias. (f) Complex exponential sequence n j e n x ) ( ) ( An arbitrary sequence can be expressed as a sum of scaled and delayed unit samples as: ) 7 ( ) 2 ( ) 1 ( ) 3 ( ) ( 7 2 1 3 n a n a n a n a n p 3 a 0 1 8 7 6 5 4 3 2 4 3 2 1 ) ( n p n 7 a 2 a 1 a Figure 2.3: An arbitrary sequence This can be written in general terms as  k k n k x n x ) ( ) ( ) ( 1.6 LINEAR SHIFT INVARIANT SYSTEM A discrete system may be thought of as a unique transformation or operator that maps some input sequence ) ( n x to some output sequence ) ( n y through some transformation ] [ T such that )] ( [ ) ( n x T n y ] [ T ) ( n x ) ( n y
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ECE 524E DIGITAL SIGNAL PROCESSING (Mr. Chemweno) 12 Figure 2.4 : Representation of a system A linear system obeys the principle of superposition which states: If )] ( [ ) ( 1 1 n x T n y and )] ( [ ) ( 2 2 n x T n y , then a system is linear if ) ( ) ( )] ( [ )] ( [ )] ( ) ( [ 2 1 2 1 2 1 n y n y n x T n x T n x n x T , are constants Example 2.1 : Determine the linearity of a 3-sample average given by )] ( [ )] 1 ( ) ( ) 1 ( [ 3 1 ) ( n x T n x n x n x n y Solution: )] ( ) ( [ 2 1 n x n x T ) 1 ( ) 1 ( ) ( ) ( ) 1 ( ) 1 ( [ 3 1 2 1 2 1 2 1 n x n x n x n x n x n x )] 1 ( ) ( ) 1 ( [ 3 )] 1 ( ) ( ) 1 ( [ 3 2 2 2 1 1 1 n x n x n x n x n x n x ) ( ) ( 2 1 n y n y Hence linear. Example 2.2 : Determine the linearity of the system ) ( )] ( [ ) ( 2 n x n x T n y Solution: )] ( ) ( [ 2 1 n x n x T 2 2 1 )] ( ) ( [ n x n x ) ( ) ( 2 ) ( ) ( 2 1 2 2 2 2 1 2 n x n x n x n x  ) ( ) ( 2 2 2 2 1 2 n x n x Hence non linear A discrete time system is shift invariant for all n and o n )] ( [ ) ( o o n n x T n n y , o n is the number of delay samples. Example 2.3 : A system is described by ) ( )] ( [ ) ( n nx n x T n y Determine whether the system is linear and whether it is time varying Solution
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ECE 524E DIGITAL SIGNAL PROCESSING (Mr. Chemweno) 13 Linearity: )] ( ) ( [ 2 1 n x n x T 2 2 1 )] ( ) ( [ n nx n nx )] ( [ )] ( [ 2 1 n nx n nx ) ( ) ( 2 1 n y n y Hence lnear Shift invariance ) ( )] ( [ ) ( o o o o n n x n n n n x T n n y Since the coefficient is time varying, then the system is not shift invariant. A system that satisfies both the above two conditions is known as Linear Shift Invariant system. 1.7 UNIT SAMPLE RESPONSE A Linear shift invariant system can be characterized by its unit sample response ) ( n h .
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