ECE 700 Lecture Notes (Phil Schniter)

ECE 700 Lecture Notes (Phil Schniter) - ECE-700 – Review...

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Unformatted text preview: ECE-700 – Review Phil Schniter January 5, 2007 1 Transforms Using x c ( t ) to denote a continuous-time signal at time t ∈ R , • Laplace Transform: X c ( s ) = integraldisplay ∞-∞ x c ( t ) e- st dt, s ∈ C • Continuous-Time Fourier Transform (CTFT): X c ( j Ω) = integraldisplay ∞-∞ x c ( t ) e- j Ω t dt, Ω ∈ R x c ( t ) = 1 2 π integraldisplay ∞-∞ X c ( j Ω) e j Ω t d Ω , t ∈ R Note that: ♣ X c ( j Ω) is the Laplace transform evaluated at s = j Ω. ♣ x c ( t ) ∈ R implies X c ( j Ω) = X * c ( − j Ω), i.e., “conjugate symmetry”. ♣ x c ( t ) is “bandlimited” to Ω if X c ( j Ω) = 0 for all | Ω | > Ω . Using x [ n ] to denote a discrete-time signal at index n ∈ Z , • z-transform: X ( z ) = ∞ summationdisplay n =-∞ x [ n ] z- n , z ∈ C Denoting a transform pair by x [ n ] ↔ X ( z ), some useful properties are x [ − n ] ↔ X ( z- 1 ) ( − 1) n x [ n ] ↔ X ( − z ) c circlecopyrt P. Schniter, 2002 1 • Discrete-Time Fourier Transform (DTFT): X ( e jω ) = ∞ summationdisplay n =-∞ x [ n ] e- jωn , ω ∈ R x [ n ] = 1 2 π integraldisplay π- π X ( e jω ) e jωn dω, n ∈ Z Note that: ♣ X ( e jω ) is the z-transform evaluated on the unit circle in C-plane: z = e jω . ♣ X ( e jω ) is 2 π-periodic in ω . ♣ x [ n ] ∈ R implies X ( e jω ) = X * ( e- jω ), i.e., “conjugate symmetry”. Other DTFT properties are: x [ − n ] ↔ X ( e- jω ) x * [ n ] ↔ X * ( e- jω ) x [ n − ℓ ] ↔ X ( e jω ) e- jωℓ x [ n ] e jω n ↔ X ( e j ( ω- ω ) ) x [ n ] y [ n ] ↔ 1 2 π integraltext π- π X ( e jθ ) Y ( e j ( ω- θ ) ) dθ x [ n ] ∗ y [ n ] ↔ X ( e jω ) Y ( e jω ) ∑ n | x [ n ] | 2 ↔ 1 2 π integraltext π- π | X ( e jω ) | 2 dω where “ ∗ ” denotes linear convolution: x [ n ] ∗ y [ n ] = ∑ ∞ m =-∞ x [ m ] y [ n − m ]. 2 Uniform Sampling Say that x [ n ] is sampled from x c ( t ) with uniform sampling interval T : x [ n ] = x c ( nT ) , n ∈ Z Let us define the continuous-time “impulse train” p ( t ) = ∞ summationdisplay n =-∞ δ ( t − nT ) where δ ( t ) denotes the Dirac delta function, defined by the properties integraltext ∞-∞ δ ( t ) dt = 1 “unit area” integraltext ∞-∞ f ( t ) δ ( t − τ ) dt = f ( τ ) “sifting property” Using Fourier series, it can be shown that p ( t ) = 1 T ∞ summationdisplay k =-∞ e j 2 π T kt c circlecopyrt P. Schniter, 2002 2 x c ( t ) x s ( t ) x [ n ] t t n Figure 1: Signals used in sampling theorem. Multiplying x c ( t ) by the impulse train yields the “continuous-time sampled signal” x s ( t ) which will help us to derive the sampling theorem. (See Fig. 1.) x s ( t ) = x c ( t ) ∞ summationdisplay n =-∞ δ ( t − nT ) = x c ( t ) 1 T ∞ summationdisplay k =-∞ e j 2 π T kt Taking the CTFT of x s ( t ), X s ( j Ω) = integraldisplay ∞-∞ parenleftBigg x c ( t ) 1 T ∞ summationdisplay k =-∞ e j 2 π T kt parenrightBigg e- j Ω t dt = 1 T...
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ECE 700 Lecture Notes (Phil Schniter) - ECE-700 – Review...

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