Bài giảng Xử lý tín hiệu số: Sampling and Reconstruction - Ngô Quốc Cường (ĐH Sư phạm Kỹ thuật)

Bài giảng "Xử lý tín hiệu số: Sampling and Reconstruction" cung cấp cho người học các kiến thức: Introduction, review of analog signal, sampling theorem, analog reconstruction. Mời các bạn cùng tham khảo nội dung chi tiết.
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Bài giảng Xử lý tín hiệu số: Sampling and Reconstruction - Ngô Quốc Cường (ĐH Sư phạm Kỹ thuật) Xử lý tín hiệu số Sampling and ReconstructionNgô Quốc Cườngngoquoccuong175@gmail.com Ngô Quốc Cườngsites.google.com/a/hcmute.edu.vn/ngoquoccuong Sampling and reconstruction• Introduction• Review of analog signal• Sampling theorem• Analog reconstruction 2 1.1 IntroductionDigital processing of analog signals proceeds in threestages:1. The analog signal is digitized, that is, it is sampled and eachsample quantized to a finite number of bits. This process iscalled A/D conversion.2. The digitized samples are processed by a digital signalprocessor.3. The resulting output samples may be converted back intoanalog form by an analog reconstructor (D/A conversion). 3 1.2 Review of analog signal• An analog signal is described by a function of time, say, x(t). The Fourier transform X(Ω) of x(t) is the frequency spectrum of the signal:• The physical meaning of X(Ω) is brought out by the inverse Fourier transform, which expresses the arbitrary signal x(t) as a linear superposition of sinusoids of different frequencies: 4 1.2 Review of analog signal• The response of a linear system to an input signal x(t):• The system is characterized completely by the impulse response function h(t). The output y(t) is obtained in the time domain by convolution: 5 1.2 Review of analog signal• In the frequency domain by multiplication:• where H(Ω) is the frequency response of the system, defined as the Fourier transform of the impulse response h(t): 61.2 Review of analog signal 7 CT Fourier Transforms of Periodic SignalsSource: Jacob White 8 Fourier Transform of CosineSource: Jacob White 9 Impulse Train (Sampling Function) Note: (period in t) T (period in ) 2/TSource: Jacob White 10 1.3 Sampling theorem• The sampling process is illustrated in Fig. 1.3.1, where the analog signal x(t) is periodically measured every T seconds. Thus, time is discretized in units of the sampling interval T: 11 1.3 Sampling theorem• For system design purposes, two questions must be answered:1. What is the effect of sampling on the original frequencyspectrum?2. How should one choose the sampling interval T? 12 1.3 Sampling theorem• Although the sampling process generates high frequency components, these components appear in a very regular fashion, that is, every frequency component of the original signal is periodically replicated over the entire frequency axis, with period given by the sampling rate:• Let x(t) = xc(t), sampling pulse s(t) 13 1.3 Sampling theoremSource: Zheng-Hua Tan 14 1.3 Sampling theoremSource: Zheng-Hua Tan 15 1.3 Sampling theoremSource: Zheng-Hua Tan 16 1.3 Sampling theoremSource: Zheng-Hua Tan 171.3 Sampling theorem Source: Zheng-Hua Tan 181.3 Sampling theorem Source: Zheng-Hua Tan 19 1.3 Sampling theorem• The sampling theorem provides a quantitative answer to the question of how to choose the sampling time interval T.• T must be small enough so that signal variations that occur between samples are not lost. But how small is small enough?• It would be very impractical to choose T too small because then there would be too many samples to be processed. 20