Presented by Huanhuan Chen University of Science and Technology of China 信号与信息处理 Signal and Information Processing

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An Introduction to Signal and Information Processing About signal Some intuitions of signals Convert analog signal to discrete signal Some elemental signals Introduction to Signal Continuous signals Discrete signals Time domain Frequency domain …… Properties of signals Convolution DFT …… The operations on the signal

About Signal The term signal is generally applied to something that conveys information. The independent variable in the mathematical representation of a signal may be either continuous or discrete.  Voltage across a resister  Velocity of a vehicle  Light intensity of an image  Temperature, pressure inside a system

1-d signals Seismic vibrations EEG and EKG Speech Sonar Audio Music ph - o - n - e - t - i - c - ia - n

2-d signals. Photographs Medical images Radar IED detection Satellite data Fax Fingerprints

3-d signals. Video Sequences Motion Sensing Volumetric data sets Computed Tomography, Synthetic Aperture Radar Reconstruction)

An bird’s view of signal processing Processing system Information Signal Recognition - radar, sonar, seismic, … Storage Transmission Processing system Display Information Communications Storage Media

Continuous-Time signal Continuous-time signals are defined along a continuum of times and thus are represented by a continuous independent variable. Continuous-time signals are often referred to as analog signals.

Discrete-time signal

Some basic sequences The sequences shown play important roles in the analysis and representation of discrete-time signals and systems.

Unit Step Function Precise GraphCommonly-Used Graph

Signum Function Precise GraphCommonly-Used Graph The signum function, is closely related to the unit-step function.

Unit Ramp Function The unit ramp function is the integral of the unit step function. It is called the unit ramp function because for positive t, its slope is one amplitude unit per time.

Rectangular Pulse or Gate Function Rectangular pulse,

Unit Impulse Function So unit impulse function is the derivative of the unit step function or unit step is the integral of the unit impulse function Functions that approach unit step and unit impulse

Representation of Impulse Function The area under an impulse is called its strength or weight. It is represented graphically by a vertical arrow. An impulse with a strength of one is called a unit impulse.

Properties of the Impulse Function The Sampling Property The Scaling Property The Replication Property g(t) ⊗ δ (t) = g (t)

Unit Impulse Train The unit impulse train is a sum of infinitely uniformly- spaced impulses and is given by

The Unit Rectangle Function The unit rectangle or gate signal can be represented as combination of two shifted unit step signals as shown

Sinc Function

Time Domain and Frequency Domain Many ways that information can be contained in a signal.  Manmade signals.  AM  FM  Single-sideband  Pulse-code modulation  Pulse-width modulation Only two ways that are common for information to be represented.  Information represented in the time domain,  Information represented in the frequency domain.

The frequency domain Frequency domain is considered indirect. Information is contained in the overall relationship between many points in the signal. By measuring the frequency, phase, and amplitude, information can be obtained about the system producing the motion.

Converting analog to digital signals Sampling is the acquisition of the values of a continuous- time signal at discrete points in time x(t) is a continuous-time signal, x[n] is a discrete-time signal

Signal sampling Nyquist sampling theorem. The lower bound of the rate at which we should sample a signal, in order to be guaranteed there is enough information to reconstruct the original signal is 2 times the maximum frequency. Now in its digital form, we can process the signal in some way..

Sampling continuous signal

Discrete Time Unit Step Function or Unit Sequence Function

Discrete Time Unit Ramp Function

Discrete Time Unit Impulse Function or Unit Pulse Sequence

Convolution of two sequence (a)–(c) The sequences x[k] and h[n− k] as a function of k for different values of n. (Only nonzero samples are shown.) (d) Corresponding output sequence as a function of n.

Properties of Convolution Commutative Property Distributive Property Associative Property

Discrete Fourier transform Convert continuous DFT to discrete DFT. Continuous version Discrete version Let  stand for (a primitive nth root of unity) We get

Smoothing of signal with DFT

Intuitions of DFT

Boxes as Sum of Cosines

The end Thank you