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Unformatted text preview: Ve451 Lecture Notes Dianguang Ma Summer 2010 Chapter 6 Sampling and Reconstruction of Signals Equivalent ContinuousTime Signal Processing System In designing the equivalent continuoustime signal processing system, we must first select the bandwidth of the signal to be processed. Once the desired frequency band is selected we can specify the sampling rate and the characteristics of the prefilter. The Prefilter The prefilter is an analog filter which has a twofold purpose. First, it ensures that the bandwidth of the signal to be sampled is limited to the desired frequency range. Another reason for using a prefilter is to reduce the additive noise and reject outofband noise. Example A speech signal with an absolute bandwidth of 3000 Hz is to be digitized. The speech signal would be prefiltered by a lowpass filter having a passband extending to 3000 kHz, a transition band of approximately 400 to 500 Hz, and a stopband above 3400 to 3500 Hz. The speech signal may be sampled at 8000 Hz and hence the folding frequency would be 4000 Hz. The Ideal A/D Converter The ideal A/D converter and the ideal D/A converter provide the interface between the continuoustime and discretetime domains. The overall system is equivalent to a continuoustime system. If ( ) i a x t s the input and ( ) is the output of the ideal A/D, we have 1 ( ) ( ) ( ) ( ) F a a s k x n x n x nT X F X F kF T ∞ =∞ = ↔ = ∑ The Ideal D/A Converter If ( ) is the input and ( ) is the output of the ideal D/A, we have ( ) ( ) ( ) where , / 2 ( ) ( / ) ( ) 0, / 2 ( ) ( ) ( ) a a a n F s a a s a a y n y t y t y n g t nT T F F g t Sa t T G F F F Y F G F Y F π ∞ =∞ = ≤ = ↔ = = ∑ The Digital Signal Processor Suppose that we are given a continuoustime LTI system defined by ˆ ˆ ( ) ( ) ( ) ( ) ( ) ( ) and we wish to determine whether there is a discretetime system ( ) such that the entire system F a a a a a a y t h t x t Y F H F X F H F = ↔ = in Figure 6.2.1 is equivalent to the continuoustime system. ( 29 ( 29 If ( ) is bandlimited to and 2 , we have ( ) ( ) / for /2. Therefore, the output of the system is given by ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ˆ To assure that ( ) a s a s a a a a a a a x t B F B X F X F T F F Y F G F Y F G F H F X F H F G F X F H F X F y t y = ≤ = = = = = ( ), we should choose the discretetime system so that ( ), / 2 ( ) ( ), ( ) 0, / 2 s a s a k s t H F F F H F H F kF H F F F ∞ =∞ ≤ = = ∑ Example 6.2.1 Simulation of an Analog Circuit Consider the analog circuit shown in Figure 6.2.4(a). Its frequency response is ( ) 1/ ( ) 1/ ( ) ( ) ( ) 1/ ( ) 1/ ( ) ( ) ( ) Clearly the impulse response ( ) is not bandlimi a a a At a a Y j C RC H X R j C j RC A j A h t Ae u t h t Ω Ω Ω = = = Ω + Ω Ω + = Ω + = ted....
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 Summer '10
 DianguangMa
 Digital Signal Processing, Signal Processing, quantization, Xq

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