# EC3400: Introduction to Digital Signal Processing

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EC3400: Introduction to Digital Signal Processing
by Roberto Cristi Professor Dept. of ECE Naval Postgraduate School Monterey, CA 93943

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

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 ...

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

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

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.

Recall: the Fourier Transform of a continuous time signal
the Discrete Time Fourier Transform of a discrete time signal

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

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

As a consequence:

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

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

Finally, put everything together and assume ideal analog filters:
ADC DAC ZOH clock LPF anti-aliasing filter reconstruction

reconstruction filter

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

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