ABSTRACT
In Chapter 6 we were able to treat a discrete-time signal as a formally continuous function by representing the sampled signal as a weighted impulse train x I ( t ) = x ( t ) P Δ t ( t ) = ∑ ℓ = − ∞ ∞ x ( t ) δ ( t − ℓ Δ t ) = ∑ ℓ = − ∞ ∞ x ( ℓ Δ t ) δ ( t − ℓ Δ t ) , https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429144752/43ba3910-79f8-4f1c-a064-557aeec25e11/content/equ7_1.tif"/> and we derived its Fourier transform in two forms (recall Theorem 6.11 and Corollary 6.13): X I ( f ) = F { x I ( t ) } = ∑ ℓ = − ∞ ∞ x ( ℓ Δ t ) e − j 2 π f ℓ Δ t ︸ F o u r i e r s e r i e s o f X I ( f ) = 1 Δ t ∑ k = − ∞ ∞ X ( f − k Δ t ) ︸ F { x ( t ) } * F { P Δ t ( t ) } , https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429144752/43ba3910-79f8-4f1c-a064-557aeec25e11/content/equ7_2.tif"/> where X(f) = ℱ{x(t)}. Note that XI (f +1/Δt) = XI (f), so XI (f) is a periodic function with period equal to the sampling rate ℝ = 1/Δt. Recall that if x(t) is band-limited with bandwidth F ≤ R, then we may extract X(f)= ℱ{x(t)} from the central period of XI (f); otherwise the shifted replicas of X (f) will overlap, and XI (f) = X (f) for f ∈ [−ℝ/2, ℝ/2].
