DIGITAL SIGNAL PROCESSING

DR. JAMEEL AHMED

ContentsIntroduction Digital Signal Processing

Representation Of a Discrete Signal

Elementary Discrete Time Signals

Comparison Between Continuous-Time & Discrete-Time Sinusoids

Sampling of Analog Signal

Relation Between Ω,F & ω, f

Introduction To DSPSignalA signal is any physical quantity that varies with time, space or any other independent variable or variables. Real-valued, Complex valued, multichannel, multi-diemnsional

ProcessingPerforming certain operations on a signal to extract some useful information

DigitalThe word digital in digital signal processing means that the processing is done either by a digital hardware or by a digital computer.

Digital Signal Processing

Digital Signal Processing is performing signal processing using digital techniques with the aid of digital hardware and/or some kind of computing device.

Digital Signal Processor is a digital computer or processor that is designed especially for signal processing applications.

Limitations of Analog Signal Processing

Accuracy limitations due to Component tolerances Undesired nonlinearities

Limited repeatability due to Tolerances Changes in environmental conditions

Temperature Vibration

Sensitivity to electrical noise Limited dynamic range for voltage and currents Inflexibility to changes Difficulty of storing information

Advantages Of Digital Signal Processing

Accuracy can be controlled by choosing word length Repeatable Sensitivity to electrical noise is minimal Dynamic range can be controlled using floating point numbers Flexibility can be achieved with software implementations Digital storage is cheap Digital information can be encrypted for security

Limitations of Digital Signal Processing

Sampling causes loss of information A/D and D/A requires mixed-signal hardware Limited speed of processors Quantization and round-off errors

Classification Of Signals

Signal

Continuous Time Signal

Discrete Time

Signal

Continuous Valued Signal

Discrete Valued Signal

Classification Of SignalsContinuous-Time Signal

Function of timeFinite or Infinite values

Discrete-Time SignalFunction of n (number of samples)Finite or Infinite values

Continuous-Valued Signal Infinite valuesFunction of time or n

Discrete-Valued Signal Finite valuesFunction of time or n

Continuous-Time Versus Discrete-Time Signal

Continuous-Time Signals (Analog signal) are defined for every value of time. It is represented as a function of time.Where asDiscrete-Time Signals are defined only at certain specific values of time. It is represented as a function of n (number of samples).

Comparison Between Continuous Time and Discrete Time Signal

Continuous Time

x(t) = A cos(Ωt +θ ) , - ∞ < t < ∞

Ω = 2π F -∞ < F < ∞

Where

A= AmplitudeΩ = Frequency (radian/

second)θ=PhaseF=cycles/second

Discrete Time

x (n) = A cos(ω n+ θ) - ∞ < n< ∞

ω =2π f -π ≤ ω ≤ π

Where

A = Amplitudeω = Frequency

(radian/sample)θ = Phasef = cycles/sample

Characteristics Of Discrete Time Sinusoid

1. A Discrete-time sinusoid is periodic if its frequency f is a rational number.

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

3. The highest rate of oscillation in a discrete-time sinusoid is attained when ω = π ( or ω = -π) or, equivalently f=1/2 (or f= -1/2).

Characteristics Of Discrete Time Sinusoid

A Discrete-time sinusoid is periodic if its frequency f is a rational number.

Characteristics Of Discrete Time Sinusoid

Discrete-time sinusoids whose frequencies are separated by an integer multiple of 2π are identical.Consider the sinusoidIt follows

Where

are indistinguishable(i.e. identical).

The sinusoids having the frequency | |> are the alias of 𝝎 𝝅the corresponding sinusoid with frequency | |< .𝝎 𝝅

Characteristics Of Discrete Time Sinusoid

The highest rate of oscillation in a discrete-time sinusoid is attained when ω = π ( or ω = -π) or, equivalently f=1/2 (or f= -1/2).

Characteristics Of Discrete Time Sinusoid

Example of Discrete Time Sinusoidal Signal

Example # 1: x (n) = A cos(ω n+ θ)Whereω = π/6 and θ=π/3f = 1/12 cycles per sample