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Uniform Sampling of a Continuous-Time Bandlimited Signal

Module by: Phil Schniter

Summary: Covers methods involved in uniform sampling of a continuous-time bandlimited signal.

Uniform Sampling

Say that xn xn is sampled from x c t x c t with uniform sampling interval TT:

∀n,n∈ℤ:xn= x c nT n n xn x c n T (1)

Let us define the continuous-time "impulse train"

pt=∑n=-∞∞δt-nT p t n δ t nT (2)
where δt δ t denotes the Dirac delta function, defined by the properties:

unit area

∫-∞∞δtdt=1 t δ t 1 (3)

sifting property

∫-∞∞ftδt-τdt=fτ t ft δ t τ fτ (4)

Using Fourier series, it can be shown that

pt=1T∑k=-∞∞ⅇⅈ2πTkt p t 1 T k 2 T k t (5)

Figure 1: Signals used in sampling theorem.
(uni_sam.png)

Multiplying x c t x c t by the impulse train yields the "continuous-time sampled signal" x s t x s t which will help us to derive the sampling theorem. (See Figure 1)

x s t= x c t∑n=-∞∞δt-nT= x c t1T∑k=-∞∞ⅇⅈ2πTkt x s t x c t n δ t nT x c t 1 T k 2 T k t (6)

Taking the CTFT of xs t xs t,

Xs ⅈΩ=∫-∞∞ xc t1T∑k=-∞∞ⅇⅈ2πTktⅇ-ⅈΩtdt=1T∑k=-∞∞∫-∞∞ xc tⅇ-ⅈΩ-2πTktdt Xs Ω t xc t 1 T k 2 T k t Ω t 1 T k t xc t Ω 2 T k t (7)
Xs ⅈΩ=1T∑k=-∞∞ Xc ⅈΩ-2πTk Xs Ω 1 T k Xc Ω 2 T k (8)

Notice that Xs ⅈΩ Xs Ω is a summation of scaled and shifted copies of Xc ⅈΩ Xc Ω . When Xc ⅈΩ Xc Ω is bandlimited to πTrads T rad s the spectral copies do not overlap, and we say there is no "aliasing." Aliasing may result when Xc ⅈΩ Xc Ω is not bandlimited to πT T . (See Figure 2 and Figure 3) The frequency πTrads T rad s , i.e., 12THz 1 2 T Hz , is often called the Nyquist frequency.

Figure 2: Example of aliasing in Xs ⅈΩ Xs Ω
(aliasing.png)
Figure 3: Example of no aliasing in Xs ⅈΩ Xs Ω
(noaliasing.png)

We now investigate the relationship between the CTFT and the DTFT:

Xⅇⅈω=∑n=-∞∞xnⅇ-ⅈωn=∑n=-∞∞ xc nTⅇ-ⅈωn=∑n=-∞∞∫-∞∞ xc tδt-nTdtⅇ-ⅈωn=∑n=-∞∞∫-∞∞ xc tδt-nTⅇ-ⅈωndt X ω n x n ω n n xc n T ω n n t xc t δ t n T ω n n t xc t δ t n T ω n (9)
where xc tδt-nTⅇ-ⅈωn xc t δ t n T ω n is nonzero ⇔ n=tT n t T
Xⅇⅈω=∑n=-∞∞∫-∞∞ xc tδt-nTⅇ-ⅈωtTdt=∫-∞∞ xc t∑n=-∞∞ⅇ-ⅈωTtdt X ω n t xc t δ t n T ω t T t xc t n δ t n T ω T t (10)
where xc t∑n=-∞∞δt-nT= xs t xc t n δ t n T xs t
Xⅇⅈω= X s ⅈωT X ω X s ω T (11)
Plugging Equation 8 into Equation 11 yields:
Xⅇⅈω=1T∑k=-∞∞ Xc ⅈω-2πkT X ω 1 T k Xc ω 2 k T (12)

Notice that the DTFT of the sampled signal xn x n is a summation of scaled, stretched, and shifted copies of the CTFT of the continuous signal xc t xc t . As implied by Equation 11, the DTFT may show evidence of aliasing when xc t xc t is not bandlimited to the Nyquist frequency. (See Figure 4 and Figure 5)

Figure 4: Example of aliasing in Xⅇⅈω X ω .
(aliasing_scaling.png)
Figure 5: Example of no aliasing in Xⅇⅈω X ω .
(noaliasing_scaling.png)

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