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Lecture 10:The Discrete-Time Sampling

Module by: Thanh Dinh Vu, Truc Pham-Dinh, Anh Tuan Hoang, Tam Huynh-Ngoc

Summary: Sampling of DT signals is an important tool for digital signal processing.

Lecture #10:
THE DISCRETE-TIME SAMPLING
Motivation:
  • Sampling of DT signals is an important tool for digital signal processing.
Outline:
  • Sampling DT signals, up-sampling, down-sampling
  • Conclusion
I. REVIEW DT PROCESSING OF CT SIGNALS
DT filtering of CT signals can be modeled as a cascade of signal transformations. The C/D converter transforms a CT signal to a DT signal, the D/C converter transforms a DT signal to a CT signal.
Figure 1
1/ Review discrete time Fourier transform (DTFT)
Figure 2
An aperiodic discrete time function has a DTFT that is periodic in φ with period φ=1.
2/ Model of C/D converter
Figure 3
3/ Model of D/C converter
Figure 4
II. SAMPLING OF DT SIGNALS
DT signals can be sampled in a manner analogous to the sampling of CT signals. We develop the Sampling Theorem for DT signals and then consider two additional sampling schemes called upsampling and downsampling. In upsampling, the time scale of the sample wave is expanded, in downsampling the time scale is compressed. These methods are integral parts of the design of digital signal processing systems.
1/ Sampling
Sampling of DT signals is analogous to sampling of CT signals.
Figure 5
The sampling theorem for DT signal follows directly from examining the Fourier transform of the sampled DT sequence.
Figure 6
No overlap of lobes occurs if
1 N − a > a 1 N − a > a size 12{ { {1} over {N} } - a>a} {}
Hence, to recover the sampled DT signal from its samples, the sampling frequency must exceed twice the highest frequency in the DT signal, i.e., 1/N > 2α.
2/ Upsampling — expanding the time scale
An important operation that is used in digital signal processing is upsampling.
Figure 7
We will determine the effect of upsampling on the Fourier transform by first showing that upsampling can be represented by
x up = ∑ m = − ∞ ∞ x [ m ] δ [ n − mN ] x up = ∑ m = − ∞ ∞ x [ m ] δ [ n − mN ] size 12{x rSub { size 8{ ital "up"} } = Sum cSub { size 8{m= - infinity } } cSup { size 8{ infinity } } {x \[ m \] δ \[ n - ital "mN" \] } } {}
What does δ[n − mN] mean? Note
δ [ n − mN ] = { 1 if n − mN = 0 and m ∧ n are int egers 0 otherwise δ [ n − mN ] = { 1 if n − mN = 0 and m ∧ n are int egers 0 otherwise size 12{δ \[ n - ital "mN" \] = left lbrace matrix { 1 matrix { {} # {} } ital "if" matrix { {} # {} } n - ital "mN"=0 matrix { {} # {} } ital "and" matrix { {} # {} } m&n matrix { {} # {} } ital "are" matrix { {} # {} } "int" ital "egers" {} ## 0 matrix { matrix { matrix { matrix { matrix { {} # {} } {} # {} } {} # {} # {} } {} # {} } {} # {} } ital "otherwise"{} } right none } {}
Therefore, δ[n − mN] = 1 for the following conditions
if m = 0 then n = 0 ,
if m = 1 then n = N ,
if m = 2 then n = 2N,
if m = 3 then n = 3N,
Figure 8
Now examine
x up = ∑ m = − ∞ ∞ x [ m ] δ [ n − mN ] x up = ∑ m = − ∞ ∞ x [ m ] δ [ n − mN ] size 12{x rSub { size 8{ ital "up"} } = Sum cSub { size 8{m= - infinity } } cSup { size 8{ infinity } } {x \[ m \] δ \[ n - ital "mN" \] } } {}
term by term as follows.
Figure 9
Now we can find the Fourier transform of xup[n]xup[n] size 12{x rSub { size 8{ ital "up"} } \[ n \] } {} as follows:
X ~ up ( ϕ ) = ∑ n = − ∞ ∞ x up [ n ] e − j2πnϕ X ~ up ( ϕ ) = ∑ n = − ∞ ∞ ∑ m = − ∞ ∞ x [ m ] δ [ n − mN ] e − j2πnϕ X ~ up ( ϕ ) = ∑ n = − ∞ ∞ x [ m ] ∑ m = − ∞ ∞ δ [ n − mN ] e − j2πnϕ X ~ up ( ϕ ) = ∑ m = − ∞ ∞ x [ m ] e − j2πnϕ X ~ up ( ϕ ) = X ~ ( Nϕ ) X ~ up ( ϕ ) = ∑ n = − ∞ ∞ x up [ n ] e − j2πnϕ X ~ up ( ϕ ) = ∑ n = − ∞ ∞ ∑ m = − ∞ ∞ x [ m ] δ [ n − mN ] e − j2πnϕ X ~ up ( ϕ ) = ∑ n = − ∞ ∞ x [ m ] ∑ m = − ∞ ∞ δ [ n − mN ] e − j2πnϕ X ~ up ( ϕ ) = ∑ m = − ∞ ∞ x [ m ] e − j2πnϕ X ~ up ( ϕ ) = X ~ ( Nϕ ) alignl { stack { size 12{ {X} cSup { size 8{ "~" } } rSub { size 8{ ital "up"} } \( ϕ \) = Sum cSub { size 8{n= - infinity } } cSup { size 8{ infinity } } {x rSub { size 8{ ital "up"} } \[ n \] e rSup { size 8{ - j2πnϕ} } } } {} # {X} cSup { size 8{ "~" } } rSub { size 8{ ital "up"} } \( ϕ \) = Sum cSub { size 8{n= - infinity } } cSup { size 8{ infinity } } { left [ Sum cSub { size 8{m= - infinity } } cSup { size 8{ infinity } } {x} \[ m \] δ \[ n - ital "mN" \] right ]e rSup { size 8{ - j2πnϕ} } } {} # {X} cSup { size 8{ "~" } } rSub { size 8{ ital "up"} } \( ϕ \) = Sum cSub { size 8{n= - infinity } } cSup { size 8{ infinity } } {x \[ m \] left [ Sum cSub { size 8{m= - infinity } } cSup { size 8{ infinity } } {δ \[ n - ital "mN" \] e rSup { size 8{ - j2πnϕ} } } right ]} {} # {X} cSup { size 8{ "~" } } rSub { size 8{ ital "up"} } \( ϕ \) = Sum cSub { size 8{m= - infinity } } cSup { size 8{ infinity } } {x \[ m \] e rSup { size 8{ - j2πnϕ} } } {} # {X} cSup { size 8{ "~" } } rSub { size 8{ ital "up"} } \( ϕ \) = {X} cSup { size 8{ "~" } } \( Nϕ \) {} } } {}
Hence, upsampling results in a compression of the frequency scale.
Figure 10
Note that expanding the time scale by N compresses the frequency scale by N.
3/ Downsampling — compressing the time scale
In downsampling by N, every NthNth size 12{N rSup { size 8{ ital "th"} } } {} sample is kept and the intervening samples are removed. We can represent downsampling by a sequence of two operations we have already analyzed —sampling and decimation which is the reverse of upsampling.
Figure 11
Figure 12
Note that compressing the time scale by N expands the frequency scale by N and divides the amplitude of the DTFT by N.
III. CONCLUSIONS
  • Expansion/compression of DT signals was interpreted as compression/ expansion of the DTFT.

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