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Digital Signal Conversion Guide

The document discusses digital conversion from continuous time signals to discrete time signals. It covers sampling, where a continuous time signal is converted to a discrete time signal by taking samples at regular time intervals. It also discusses discrete-to-continuous conversion, where a discrete time signal can be reconstructed to a continuous time signal using various interpolation methods. The document provides examples of sampling sinusoids and discusses oversampling.

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43 views12 pages

Digital Signal Conversion Guide

The document discusses digital conversion from continuous time signals to discrete time signals. It covers sampling, where a continuous time signal is converted to a discrete time signal by taking samples at regular time intervals. It also discusses discrete-to-continuous conversion, where a discrete time signal can be reconstructed to a continuous time signal using various interpolation methods. The document provides examples of sampling sinusoids and discusses oversampling.

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김민성
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© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
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Chap. 4 : Digital Conversion

4.1. Sampling : Conversion from Continuous Time(CT) to Discrete Time(DT)

Continuous time signal


x(t) → to → x[n] = {x(nTs )}
Discrete time signal
↑ Ts = 1/fs ( Sampling time=1/sampling freq)

Fig. 1. Conversion from continous time(CT) signal to discrete time (DT) signal
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∗ Note that

• Time-index : n = 0, ±1, ±2, ±3, . . .


⇒ n is integer and cannot be 0.1, 0.5, 0.7, 1.8, . . . .
⇒ x[0.5], x[0.7], x[0.8] are NOT possible, i.e., NOT definable.

• EX : x(t) = cos(2πt), Ts :Sampling time


when Ts = 1 sec ⇒ x1 [n] cos(2πnTs ) = (1)n = 1
when Ts = 1/2sec ⇒ x2 [n] cos(2πn1/2) = (−1)n
when Ts = 1/3sec ⇒ x3 [n] cos(2πn1/3s)
when Ts = 1/π sec ⇒ x4 [n] cos(2πn1/π) = cos(2n)
⇒ x[n] = x1 [n] + x2 [n] is NOT possible.

Fig. 2. Discrete signal with different sampling times


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• Sampled continuous signal with various sampling interval Ts

- For the fasten varying signals, the sampling interval should be more de-
creased. in other words, sampling Time(Ts ) should be inversely propor-
tional to signal freq.

Shannon Sampling Theorem (Nyquist Theorem)


A continuous-time signal x(t) with frequencies no higher than fmax can be
reconstructed exactly from its samples x[n] = x(nTs ), if the samples are
1
taken at a rate fs = 1/Ts that is greater than 2fmax , that is, fs = Ts > 2fmax .
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• Corresponding discrete Signals

* Different sampling times


⇒ Different scales.
⇒ Maps with different scales can not be put into together.
⇒ x1 [n] ± x2 [n] are NOT possible!
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• Sampling sinusoids

x(t) = A cos(ω0 t + φ)

⇓ Sampling with sampling time Ts

{x(nTs )} = {A cos(ωnTs + φ)}

x[n] = A cos(ω̂n + φ) = Re{Aejφ · ej ω̂n }

Normalized freq. : ω̂ = ωTs = ω/fs


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4.2 Discrete-to-Continuous Conversion

Discrete time signal


y[n] −→ to −→ y(t)
Continuous time signal

∞
X
y(t) = y[n]p(t − nTs ), where p(t) is an interpolator.
n=−∞

• Various interpolators p(t)

⇒ Zero crossing occurs at every sampling point.


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 1 : |t| < T /2
s
• Zero-order hold p(t) =
 0 : Otherwise


 1 − |t|/T : t
s
• First order linear interpolator p(t) =
0 : Otherwise
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• Parabolic Interpolator p(t) : 2n d order polynomial with p(t) = 0 for


t = 0, ±Ts , ±2Ts

• Ideal Band limited Interpolator


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• Oversampling
- Oversampling : sampling freq. is much greather than the Nyquist freq.
- With oversampling, the simples interpolator may reconstruct the continu-
ous signal.
- Ex : CD player, fmax ≈ 20 KHz ⇒ 4 or 3 times oversampling
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