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DIGITAL SIGNAL PROCESSING
DR. JAMEEL AHMED
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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
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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.
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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.
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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
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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
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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
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Classification Of Signals
Signal
Continuous Time Signal
Discrete Time
Signal
Continuous Valued Signal
Discrete Valued Signal
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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
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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).
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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
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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).
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Characteristics Of Discrete Time Sinusoid
A Discrete-time sinusoid is periodic if its frequency f is a rational number.
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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
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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).
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Characteristics Of Discrete Time Sinusoid
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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