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Introduction to DSP
Source: http://www.dspguide.com Free textbook on DSP
1 Version 2
Digital Signal Processing
• Key aspect is signals • Signals acquired from the environment
through sensors • Digitized through ADCs • DSP answers “what next”?
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DSP Roots
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DSP overlaps with many fields…
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Signals and Systems
• Signal – Description of how one parameter varies with another parameter – Voltage vs. time – Brightness vs. distance
• System – Any process that produces output signal in response to an input signal
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Continuous and Discrete systems
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Understanding a System
• We need to understand a system i.e., how it transforms the input signal to generate an output signal
• Examples: – How to remove noise from an ECG – How does telephone line transmit your voice
signal and changes along the way… • Too many systems with varied characteristics
– Is it really possible?
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Linear System
• Fortunately, many useful systems are linear in nature
• They possess properties that make them amenable to be abstracted and studied
• What properties make a system linear?
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Linear System
• A linear system satisfies two mathematical properties: – Homogeneity – Additivity
• If we show that a system exhibits the above two properties, then we prove that the system is linear
• A third property known as shift invariance is not required for linear system but usually satisfied
• It is mandatory of linear systems for DSP
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Homogeneity
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Additivity
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Shift Invariance
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Additional Properties
• Unfortunately, the three properties of linear systems, namely, homogeneity, additivity, and shift invariance are not enough to understand/study linear systems
• Two additional properties usually displayed by linear systems – Static linearity – Sinusoidal fidelity
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Static Linearity • defines how a linear system reacts when the signals aren't changing, i.e.,
when they are DC or static. • static response of a linear system is very simple: the output is the input
multiplied by a constant. • All linear systems have static linearity, the opposite is not always true
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Examples of Static nonlinearity
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Sinusoidal Fidelity
• Important characteristic exhibited by linear systems
• If the input to a linear system is a sinusoidal wave, the output will also be a sinusoidal wave, and at exactly the same frequency as the input.
• However, the output may differ in amplitude and phase
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Examples of Linear Systems
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Examples of Non-linear systems
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Special Properties of Linearity
• Linearity is commutative
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Special Properties of Linearity
• A system with multiple inputs and/or outputs will be linear if it is composed of linear subsystems and additions of signals. The complexity does not matter, only that nothing nonlinear is allowed inside of the system.
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Multiply Operation
• With a constant is linear… • Multiplication of two signals is non-linear
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Synthesis & Decomposition
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Superposition – Foundation
of DSP
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Common Decompositions
• Using Impulses Impulse=single non zero
value in a sequence of zeroes
δ[n] δ[n-s]
• Using Step signals
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0 s
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Fourier Decomposition
• Not very obvious • N-point signal decomposed to N+2 signals • Half are sine waves • Half are cosine waves
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Fourier Decomposition
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Convolution
• Convolution is a mathematical way of combining two signals to form a third signal
• The most important concept in DSP • Input and impulse response are combined to
compute the output signal
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Impulse Response of a Linear System
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Computing output signal
Question: If a linear system has an impulse response of h[n] then compute the output for -3*δ[n-8]
Answer: If δ[n] gives output h[n] then δ[n-8] gives output h[n-8] … shift invariant
and -3* δ[n-8] gives output 3*h[n-8] … homogeneity
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Computing Output Response
Given an input signal, we can compute the output of a linear system if we know its impulse response
Step 1: Decompose signal into sum of scaled and shifted impulses
Step 2: Compute output for each scaled and shifted impulse input
Step 3: Add all the output signals
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Convolution
• Is another mathematical operation just like add, subtract, multiply
• Represented by * (asterisk) • y[n] = x[n] * h[n] • “x[n] is convolved with h[n] to produce y[n]”
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Filtering Example
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Input = Sine wave
(High Frequency) +
Ramp (Low Frequency)
More Examples
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Input and Output Lengths • Output Length = Input Length + Impulse Length – 1
• Usually Impulse length is very short (~100) • While input can be very large (~millions of samples)
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= 81 + 31 – 1 = 111
= 81 + 31 – 1 = 111
Understanding Convolution
• We can understand from the perspective of – Input Signal (Input side algorithm) – Output Signal (Output side algorithm)
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Input Side Algorithm
• This first viewpoint of convolution is based on the fundamental concept of DSP – decompose the input, – pass the components through the system, – and synthesize the output.
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Input Side Algorithm – An illustrative example
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Same example – input and impulse response swapped
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We observe that we get the
same output
Convolution Is commutative
Output Side Algorithm
• Input side perspective answers the following question – How does each input sample contribute to the
output? – This approach is not a natural way of computing
output • Output side perspective tries to answer the
following question – For each output sample, what input samples
contribute to its generation? – This approach is more natural
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Example revisited
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Which inputs contribute to output y[6]? To answer this, we can look at sample 6 in each plot We see that X[3], x[4], x[5], and x[6] contribute to y[6] i.e., y[6]=x[3]h[3] + x[4]h[2] + x[5]h[1] + x[6]h[0]
Example revisited
• y[6] = x[3]h[3] + x[4]h[2] + x[5]h[1] + x[6]h[0] • Note that samples of h in the above equation
are in decreasing order • While input samples are in an increasing order • From the output perspective, thus, the
impulse response is “flipped”
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Samples Not Available!!
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Samples Not Available!!
Convolution – Mathematical Formula
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End effects of convolution
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Convolution - Applications
• A linear system's characteristics are completely specified by the system's impulse response, as governed by the mathematics of convolution
• The above is the basis for many signal processing techniques
• Examples – Filters – systems with appropriate impulse response – Enemy aircraft detected by radar by analyzing measured
impulse response – Echo suppression in long distance phone calls
• Create impulse response that counteracts impulse response of echo
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Example Low Pass Filter Kernels
• Kernel another word for Impulse Response
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High Pass Filter Kernels
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