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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. DSP Hardware Software sonar audio radar video... filtered signal: reject disturbances. transformed signal: detection compression A signal can come from a number of different sources:

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ADC DSP DAC LPF antialiasing LPF reconstruction LPF ADC DSP DAC LPF A Digital Filter

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LPF DAC LPF We review the relations between the spectra of the signals in the following operations: Sampling: Digital Filtering: Reconstruction:

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Structure of a Digital Filter ADCDAC ZOH clock Problem: determine the continuous time frequency response. LPF anti-aliasing filter reconstruction filter continuous timediscrete timecontinuous time

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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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ADC mathematical model of the sampler: it appends a to each sample Sampling of a continuous time signal:

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FT since FT since We can write the same expression in two different ways:

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As a consequence:

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Particular case: if the signal is bandlimited as then Notice: F is in Hz (1/sec), is in radians/sample (no dimension). LPF

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DAC ZOH Reconstruction : the Zero Order Hold 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: ADCDAC ZOH clock LPF anti-aliasing filter reconstruction filter

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reconstruction filter

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Example: suppose we design a notch discrete time filter with transfer function z-plane with zeros and poles and sampling frequency. Determine the magnitude of its frequency response in the continuous time domain.

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

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