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**EC3400: Introduction to Digital Signal Processing**

by Roberto Cristi Professor Dept. of ECE Naval Postgraduate School Monterey, CA 93943

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Week 1 Topics: Introduction Fourier Transform (Review) Sampling Reconstruction Digital Filtering Example: a Digital Notch Filter

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Introduction Objectives In this course we introduce techniques to process signals by digital computers. A signal can come from a number of different sources: filtered signal: reject disturbances. sonar DSP Hardware Software radar transformed signal: detection compression audio video ...

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**A Digital Filter ADC LPF DSP LPF LPF ADC DSP LPF antialiasing**

DAC LPF LPF ADC DSP DAC LPF antialiasing reconstruction

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**We review the relations between the spectra of the signals in the following operations:**

Sampling: LPF Digital Filtering: DAC LPF Reconstruction:

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**Structure of a Digital Filter**

continuous time discrete time continuous time ADC DAC LPF ZOH LPF anti-aliasing filter reconstruction filter clock Problem: determine the continuous time frequency response.

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**Recall: the Fourier Transform of a continuous time signal**

the Discrete Time Fourier Transform of a discrete time signal

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**Sampling of a continuous time signal:**

ADC mathematical model of the sampler: it appends a to each sample

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**We can write the same expression in two different ways:**

FT FT since since

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As a consequence:

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**Particular case: if the signal is bandlimited as**

then LPF Notice: F is in Hz (1/sec), is in radians/sample (no dimension).

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**Reconstruction: the Zero Order Hold**

DAC ZOH where g(t) is the pulse associated to each sample. Then, its FT is computed as: where G(F)=FT[g(t)] is given by

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**Finally, put everything together and assume ideal analog filters:**

ADC DAC ZOH clock LPF anti-aliasing filter reconstruction

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**reconstruction filter**

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**Example: suppose we design a notch discrete time filter with transfer function**

with zeros and poles and sampling frequency Determine the magnitude of its frequency response in the continuous time domain. z-plane

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**Solution: from what we have seen the frequency response is given by**

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