Digital Signal Processing · video signals are analog. Digital Signal Processing - Lecture 2 17 Digital signal processing system •To convert analog signals into a sequence of numbers

Digital Signal Processing

Soma Biswas

Department of Electrical Engineering

IISc, Bangalore

Office: C 320 (EE)E-mail: [email protected]

Class Hours: Tuesday and Thursday 3:30 to 5.00 pm

Signals

Digital Signal Processing - Lecture 2 2

• Signal: A physical quantity that varies with time, space, or any other independent variable or variables.

• Mathematically, we describe a signal as a function of one or more independent variables.

𝒇𝟏 𝒕 = 𝟐𝒕 𝒇𝟐 𝒕 = 𝟓𝒕𝟐

𝒇𝟑 𝒙, 𝒚 = 𝟐𝒙𝟐 + 𝟑𝒙𝒚 + 𝟓𝒚𝟐,𝒙, 𝒚 𝒄𝒐𝒖𝒍𝒅 𝒃𝒆 𝒕𝒘𝒐 𝒔𝒑𝒂𝒕𝒊𝒂𝒍 𝒄𝒐𝒐𝒓𝒅𝒊𝒏𝒂𝒕𝒆𝒔

• However, there are cases where such functional relationships are unknown or too complicated to be of any practical use.

Speech Signals


• Cannot be described functionally using such expressions.

• Generated by forcing air through the vocal cord.

ECG (Electrocardiogram)


• ECG provides doctor with information about the condition of the patient’s heart.

EEG (Electroencephalogram)


• ECG provides doctor with information about the activity of the brain.

1-D, 2-D Signal


1D 2D

Lenna or Lena (1973). originally cropped fromNov 1972 issue of Playboy magazine. It is apicture of Lena Söderberg, a Swedish model, shotby photographer Dwight Hooker.

- Image – function of 2 independent variables

- Obtained by exposing a photographic film to ascene or an object.

-Speech, ECG, EEG all are 1D Signals

- Functions of a single independent variable - time

Basic Definitions

• Amplitude: The value of the signal at a specified value of the independent variable.

• Waveform: The variation of the amplitude as a function of the independent variable.

• A continuous-time signal with a continuous amplitude is

called an analog signal. E.g., speech

• A continuous-time signal with discrete-valued amplitude

is referred to as quantized boxcar signal.


Analog and Digital Signals

Analog and digital signal

• A discrete-time signal with continuous-valued amplitude is known as sampled-data signal.

• A discrete-time signal with discrete-valued amplitude represented by finite number of digits is called digital signal.


Digital Processing Systems


Analog signal processing

Digital signal processing system

• Analog signal scan be processed directly by analog systems for changing their characteristics or extracting some desired information.

• Alternate method for processing analog systems: digital signal processing

Advantages of Digital Signal Processing

• Flexibility in reconfiguring the digital signal processing operations simply by changing the program – eg. Programmable digital computer

• Reconfiguration of analog systems usually requires redesign of hardware followed by testing and verification to see that it operates properly.

• Better control in accuracy requirements – tolerances in analog circuit components make it difficult to control the accuracy of an analog signal processing system

• Signals can be easily stored in media (like tape or disk) – so signals become transportable and can be processed offline.

• Complicated mathematical operations are easy to do on digital signal compared to analog.

• Some cases digital is cheaper – digital hardware is cheaper and flexibility of modifications

• So DSP is widely used – eg in speech processing, signal transmission on telephone channels, image processing, etc.


Limitations

• Speed of operation of A/D converters and digital signal processors.

• Extremely high bandwidth signal requires fast-sampling-rate A/D converters and fast digital signal processors.

• There are analog signals with large bandwidths for which a digital processing approach is beyond the state of the art of digital hardware.


Concept of Frequency

• Subscript a denotes x(t) is an analog signal.


𝒙𝒂 𝒕 = 𝑨𝒄𝒐𝒔 𝛀𝒕 + 𝜽 ,−∞ < 𝒕 < ∞

𝒙𝒂 𝒕 = 𝑨𝒄𝒐𝒔 𝟐𝝅𝑭𝒕 + 𝜽 ,−∞ < 𝒕 < ∞

Properties of sinusoid


•

Discrete-time Sinusoid Signals


𝒙 𝒏 = 𝑨𝒄𝒐𝒔 𝝎𝒏 + 𝜽 ,−∞ < 𝒏 < ∞

𝒙 𝒏 = 𝑨𝒄𝒐𝒔 𝟐𝝅𝒇𝒏 + 𝜽 ,−∞ < 𝒏 < ∞

𝜔

=𝜋

6𝑟𝑎𝑑𝑖𝑎𝑛 𝑝𝑒𝑟 𝑠𝑎𝑚𝑝𝑙𝑒 𝑓

Properties of discrete-time sinusoid

• A discrete-time sinusoid is periodic only if its frequency 𝒇 is a rational number. 𝒙 𝒏 + 𝑵 = 𝒙 𝒏 , 𝒇𝒐𝒓 𝒂𝒍𝒍 𝒏. The smallest value of 𝑵(> 𝟎) for which it is true is called fundamental period.

• Discrete-time sinusoids whose frequencies are separated by an integer multiple of 𝟐𝝅 are identical.


Therefore, all sinusoidal sequences, 𝑥𝑘 𝑛 = 𝐴𝑐𝑜𝑠 𝜔𝑘𝑛 + 𝜃 , 𝑘 = 0,1,2, … where, 𝜔𝑘 = 𝜔0 + 2𝜋𝑘,−𝜋 ≤ 𝜔0 ≤ 𝜋 are indistinguishable(i.e., identical).

Any sequence resulting from a sinusoid with a frequency 𝜔 > 𝜋, or 𝑓 >1

2, is

identical to a sequence obtained from a sinusoidal signal with frequency 𝜔 < 𝜋.We call the sinusoid having the frequency 𝜔 > 𝜋 an aliasof a corresponding sinusoid with frequency 𝜔 < 𝜋.

−𝜋 ≤ 𝜔 ≤ 𝜋, or −1

2≤ 𝑓 ≤

1

2, unique and all frequencies 𝜔 > 𝜋, or 𝑓 >

1

2, an alias

Properties cont…

• The highest rate of oscillation in a discrete-time sinusoid is

attained when 𝜔 = ±𝜋 or equivalently 𝑓 = ±1

2.


Analog-to-Digital and Digital-to-Analog Conversion

• Most signals of practical interest, such as speech, biological signals, seismic signals, radar signals, sonar signals, and various communication signals such as audio and video signals are analog.


Digital signal processing system

• To convert analog signals into a sequence of numbers having finite precision, we use analog-to-digital (A/D) conversion, and the corresponding devices are called A/D Converters (ADCs).

A/D converter

• Sampling: Takes “samples” at discrete time interval. 𝒙 𝒏 =𝒙𝒂 𝒏𝑻 ,𝒘𝒉𝒆𝒓𝒆 𝑻 𝒊𝒔 𝒄𝒂𝒍𝒍𝒆𝒅 𝒔𝒂𝒎𝒑𝒍𝒊𝒏𝒈 𝒊𝒏𝒕𝒆𝒓𝒗𝒂𝒍.

• Quantization: converts discrete-time continuous-valued signal into discrete-time discrete-valued signal. Difference between 𝒙 𝒏 and 𝒙𝒒 𝒏 is called quantization error.

• Coding: Each discrete value of 𝒙𝒒 𝒏 is represented by b-bit binary sequence.


D/A converter

• In many practical cases (e.g., speech processing) it is desirable to convert the processed digital signals into analog form.

• The process of converting digital signal into an analog signal is called digital-to-analog (D/A) conversion.

• D/A converters do some form of interpolation

• Zero-order hold/staircase approximation

- Linear interpolation- linearly connect pair of

successive samples

- Quadratic interpolation, etc.


• Sampling does not result in a loss of information, nor does it introduces distortion in the signal if the signal bandwidth is finite.

• On the other hand quantization is a noninvertible or irreversible process that results in signal distortion.

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Digital Signal Processing · video signals are analog. Digital Signal Processing - Lecture 2 17 Digital signal processing system •To convert analog signals into a sequence of numbers