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Signals and Systems, Midterm Exam Solutions Spring 2004, Edited by bypeng 1. (5) A triangular pulse signal (2 4) xt + is depicted in Fig. P1. Sketch the signal (3 ) (3 2) xt xt + + . Fig. P1 Solution: 4) (2( 2)) x t += + , (2 ) x t and then () x t are depicted in Fig. P1-1(a) and (b), respectively: (a) (2 ) x t Fig. P1-1 (b) () x t Thus (3 ) x t and (3 2) + can be sketched as Fig. P1-2(a) and P1-2(b), respectively: (a) (3 ) x t Fig. P1-2 (3 2) + Therefore (3 ) (3 2) ++ is given by Fig. P1-3. Fig. P1-3

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2. (3) Determine and sketch the even and odd parts of the signal shown in Fig. P2. Label your sketch carefully. Fig. P2 Solution: () ( ) {() } 2 x tx t Ev x t +− = and } 2 x t Od x t −− = . Therefore the sketches are given as Fig. P2-1. (a) even part (b) odd part Fig. P2-1 3. (10) Consider the following continuous-time system: 0( ) 0 ( 2 ) () 0 xt yt < = + −≥ Determine whether the system is memoryless, time-invariant, linear, causal, stable? Justify your answers. Solution: Memorylessness: FALSE, since depends not only on () x t but on (2 ) . Time invariance: TRUE. 0 0 00 0 ) 0 ( 2 ) 0 xt t yt t x t t < −= −+ −− , trivially time-invariant. Linearity: FALSE. Try 1 x tt = and 2 2 x =+ , or any other possible input pairs. (NOTE: you cannot explain it in such a brief way .) Causalty: TRUE, since depends only on () x t and ) , but not on 0 x + 0 0 t > . Stability: TRUE. x t , if B ∃∈ R such that ,( ) t B ∈< R , then ) 2 ty t B ∀∈ < R . 4. (5) Consider three systems with the following input-output relationships: 1 2 11 24 even S1 : [ ] 0o d d S2 : [ ] [ ] [ 1] [ 2] S3 : [ ] [2 ] xn n yn n xn x n ⎡⎤ ⎣⎦ = = = Suppose that these systems are connected in series as follows. Find the input-output relationship (between [] in and [] on ) or the overall interconnected system.
Solution: N N N always even always odd always even [] [ 2] 11 [ 1 ] [ 2 ] 24 [ 2 ] [2 1 ] [2 2 ] 211 2 2 0 22 4 2 1 ] 4 on m n mn kn k n k n nn ii in = =+ + + ⎡⎤ + ⎢⎥ ⎣⎦ Note that neither S1 nor S3 is a time-invariant system, so one cannot change the order S1-S2-S3. 5. (6) Consifer the following input-output relationship: 3 0 1 [ ] 4 k yn xn k = = (a) (2 pt) Find the impulse response [] hn of this system. (b) (4 pt) Determine the output of the system when the input is the rectangular pulse defined as: [] [ 1 0 ] xn un un = −− Solution: (a) 3 0 1 03 [ ] [ ] ( [ ][1 ][2 ][3 ] ) 4 44 0 k n n k n n n n otherwise δδ δ = =− = + + + = (b) 9 0 01 1 13 4 [ ] [ ] 9 13 91 3 4 3 kk n n n xkhn k hn k n n n n =−∞ = ≤− +

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92ä¸‹ä¿¡è™Ÿèˆ‡ç³»çµ±

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