By Lehmer D. N.

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2 Periodicity of sinusoidal sequences While the period of the sinusoid x(t) = Dα cos(2πfα t − φα ) is always T = 1/fα , we cannot say the same for its sampled sequence for two reasons: 1. The discrete-time sample sequence may or may not be periodic depending on the sampling interval t; 2. If the discrete-time sample sequence is periodic, its period varies with the sampling interval t. To nd out whether a discrete-time sinusoid is periodic and to determine the period (measured by the number of samples), we make use of the mathematical expression for the th sample, namely, x = Dα cos(2πFα − φα ), = 0, 1, 2, .

2 2j Examples of future use: n • Prove ejkθ = sin k=−n π • Prove • Prove • Prove 1 2 sin n + θ 2 θ . 2, page 84) n ejkθ dθ = 2π. 2, page 85) −π k=−n π sin n + −π θ 2 1 2fc sin fc −fc 1 2 θ dθ = 2π. 2, page 85) sin 2πfc t . 7. REVIEW OF RESULTS AND TECHNIQUES 15 Technique 2 Trigonometric identities and their alternate forms: cos(α ± β) = cos α cos β ∓ sin α sin β, sin(α ± β) = sin α cos β ± cos α sin β, cos(α + β) + cos(α − β) , 2 cos(α − β) − cos(α + β) sin α sin β = , 2 cos α cos β = sin(α + β) + sin(α − β) , 2 sin(α + β) − sin(α − β) cos α sin β = .

7) given below, by which we can transform the sequence of discrete samples {x0 , x1 , . . , xN −1 } to the sequence of coef cients {X0 , X1 , . . , XN −1 } without solving a system of equations. 1. 7) N −1 −r x ωN , for r = 0, 1, . . , N − 1. 5). Since X±k are the coef cients of the complex exponential modes e±j2πkt/T , the corresponding frequencies ±fk = ±k/T are marked on the frequency grid. 2 Equally-spaced samples and computed DFT coef cients. 3 t 2 Frequency Grid (∆f = 1/T) Y ±k 1 −f5 0 −f5 0 f5 0 f5 = 5/T We defer the matrix formulation of the DFT until Chapter 4.