This document provides an overview of signals and systems concepts for a summer school on real-time applications. It defines what signals and spectra are, discusses analog versus digital signals, time sampling and quantization, system properties, and considerations for real-time systems including constraints, specifications, hardware components, and trade-offs between factors like time, energy, performance and more. Examples are provided on representing signals in the time and frequency domains.

1. Signal and Systems
REAL-TIME APPLICATIONS – GENERAL CONSIDERATIONS
SUMMER SCHOOL

2. Objectives
Signal and spectrum definitions
Analog vs Digital
Time Sampling
Quantization
Systems definition and properties
Real-Time Constrains

3. What is a signal?
Signal – a function that carry information in time and frequency

4. What is a spectrum?
The representation of a signal in frequency domain

5. Example
Example
We have a signal f(t) = 4*sin(10*t) + 3*sin(20*t) + sin(50*t) + 0.5*sin(70*t)
What is the representation of the signal in time domain?
What is the representation of the signal in frequency domain?

6. Example’s plots

8. Analog vs Digital
vs

9. Time sampling (Discrete-Time Signal)
Fs -> Sampling frequency
Ts -> Sampling period
Fs = 1/Ts
Nyquist – Shannon theorem

10. Discrete-Time Signal

11. Nyquist-Shannon Theorem
Let it be a signal with limited bandwidth (B), i.e. there is no frequency higher than B Hz in that
analog signal.
Thus, the sampling frequency must be greater or equal then B
Fs > 2B

12. Why we should care about Nyquist-
Shannon Theorem?

13. Aliasing

14. Quantization (Digital Signal)

15. Systems
Definition
Properties:
◦ Time-Invariant
◦ Causal
◦ Linear

16. Real-Time Applications
Objectives
Real-Time Application Diagram
◦ Process -> Sensor -> Signal Processing -> Command -> Process
Specifications
◦ Systems performances
◦ Energy constrains
◦ Dimension Requirements
◦ Economical Factors
Hardware components
Trade-off

17. Trade-off
Sampling period
Time vs Energy
Energy vs Performance
Acquisition vs Processing
Communication

18. Questions?