Bài giảng xử lý tín hiệu số sampling and reconstruction ngô quốc cường

The resulting output samples may be converted back into analog form by an analog reconstructor D/A conversion... 1.2 Review of analog signal • An analog signal is described by a function

Xử lý tín hiệu số Sampling and Reconstruction

Ngô Quốc Cường

Ngô Quốc Cường

ngoquoccuong175@gmail.com

sites.google.com/a/hcmute.edu.vn/ngoquoccuong

Sampling and reconstruction

• Introduction

• Review of analog signal

• Sampling theorem

• Analog reconstruction

2 The digitized samples are processed by a digital signal processor

3 The resulting output samples may be converted back into analog form by an analog reconstructor (D/A conversion)

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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:

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:

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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):

1.2 Review of analog signal

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CT Fourier Transforms of Periodic Signals

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Fourier Transform of Cosine

Source: Jacob White

Note: (period in t) T

(period in  ) 2/T Impulse Train (Sampling Function)

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:

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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:

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1.3 Sampling theorem

1.3 Sampling theorem

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Source: Zheng-Hua Tan

1.3 Sampling theorem

1.3 Sampling theorem

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Source: Zheng-Hua Tan

1.3 Sampling theorem

1.3 Sampling theorem

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Source: Zheng-Hua Tan

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

1.3 Sampling theorem

Hardware limits

• In real-time applications, each input sample must be acquired, quantized, and processed by the DSP, and the output sample converted back into analog format Many of these operations can be pipelined to reduce the total processing time

• In any case, there is a total processing or computation time,

where:

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1.4 Aliasing and reconstructor

• The set of frequencies,

are equivalent to each other

• Among the frequencies in the replicated set, there is a unique one that lies within the Nyquist interval It is obtained by

1.4 Aliasing and reconstructor

• Antialiasing prefilters

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1.4 Aliasing and reconstructor

• An ideal analog reconstructor extracts from a sampled signal all the frequency components that lie within the Nyquist interval

Exercises 1

• Let x(t) be the sum of sinusoidal signals

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Exercises 2a

• A sound wave has the form

where t is in milliseconds What is the frequency content of this signal? Which parts of it are audible and why?

Exercises 2b

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