Introduction to DSP

Source: http://www.dspguide.com Free textbook on DSP

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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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