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102_1_discussion1

# 102_1_discussion1 - ± 1 if n = 0 otherwise 1 2 4-2-4 1...

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EE102 Discussion 1 2010-09-28 vasiliy karasev 1. Review of Complex Numbers Cartesian form: Let z = a + ib , < ( z ) = a , = ( z ) = b Conjugation: z * = a - ib Magnitude: | z | = a 2 + b 2 Phase: φ ( z ) = tan - 1 ( b/a ) Polar form: Let z = re , where e = cos( φ ) + i sin( φ ) (Euler’s formula) Conjugation: z * = re - Magnitude: | z | = r Phase: φ Real part: < ( z ) = r cos( φ ) Imaginary part: = ( z ) = r sin( φ ) a) zz * = b) 1 2 ( z + z * ) = c) 1 2 ( z - z * ) = d) Given w = 1 + i , find a set of complex numbers z that satisfy 1 / 2 ≤ | z - w | ≤ 1. Sketch the region in a complex plane. e) Find a set of complex numbers z that satisfy | z - i | = | z + i | . Sketch the region in a complex plane. 2. Systems and Signals Signal - function of independent variable (e.g. x ( t )), or a sequence of values (e.g. x [ n ]) System - function of functions (i.e. signals). We can perform various operations on signals: Change amplitude: ˆ x ( t ) = 10 x ( t ) Delay: ˆ x ( t ) = 10 x ( t - 10) (Note negative sign!) Stretch in time: ˆ x ( t ) = x ( t/ 2) Combine with other signals: ˆ x ( t ) = x 1 ( t/ 2) x 2 ( t ) + x 3 ( t ) a) Given δ [ n ] shown and defined below, sketch δ

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Unformatted text preview: ± 1 if n = 0 otherwise 1 2 4-2-4 1 3-1-3 n δ[n] 1 b) Consider the periodic signal x ( t ). What is the period? Sketch ˆ x ( t ) = x (8 t ). What is the period of ˆ x ( t )? Sketch ˜ x ( t ) = ˆ x ( t ) x ( t ). What is its period? 2 4-2-4 1 3-1-3 t x(t) 1 ... ... c) Consider tri ( t ) shown and deﬁned below. Sketch tri ( t ) + 2 tri ( t-1) + 2 tri ( t-2) + tri ( t-3) tri ( t ) = ± 1- | t | if | t | ≤ 1 otherwise 1 2-1-2 t tri(t) 1 3. Bonus 1. Verify useful identities: cos( φ ) = 1 2 ( e iφ + e-iφ ) sin( φ ) = 1 2 i ( e iφ-e-iφ ) 2. Verify: e iφ = cos( φ ) + i sin( φ ) using power series. Recall: e x = ∑ ∞ n =0 x n n ! sin( φ ) = φ-φ 3 3! + φ 5 5!-φ 7 7! + φ 9 9! ... cos( φ ) = 1-φ 2 2! + φ 4 4!-φ 6 6! + φ 8 8! ... 2...
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