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

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

3. 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
...

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

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

6. 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.

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

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

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

10. As a consequence:

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

12. 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

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

14. reconstruction filter

15. 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

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