Summary1 - n = A cos ω n θ = A cos ω n 2 kπn θ The...

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Summary 1 Hamid Jafarkhani Digital Signal Processing Summary 1 – p. 1/
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Continuous-Time Sinusoidal Signals X a ( t ) = A cos(Ω t + θ ) = A cos(2 πFt + θ ) If we fix the frequency, F , then X a ( t ) is periodic. If two sinusoids have different values of F , then they are different functions. As F increases, the sinusoid oscillates faster. Summary 1 – p. 2/
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Discrete-Time Sinusoidal Signals X ( n ) = A cos( ωn + θ ) = A cos(2 πfn + θ ) A discrete-time sinusoid is periodic only if its frequency f is a rational number. X ( n ) = cos( n ) is not periodic! Discrete-time sinusoids with different frequencies can be identical. X k ( n ) = A cos( ω 0 n + θ ) =
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Unformatted text preview: n ) = A cos( ω n + θ ) = A cos( ω n + 2 kπn + θ ) The highest rate of oscillation in a discrete-time sinusoid is attained when ω = π (or ω =-π ) or equivalently f = 1 2 (or f =-1 2 ). Summary 1 – p. 3/ 4 Some Useful Definitions Energy: E = ∞ s n =-∞ | x ( n ) | 2 Cross correlation between real signals x ( n ) and y ( n ) : r xy ( l ) = ∞ s n =-∞ x ( n ) y ( n-l ) Auto correlation (real signal): r xx ( l ) = ∞ s n =-∞ x ( n ) x ( n-l ) Summary 1 – p. 4/ 4...
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