# Brief Notes + Equations

This is just a collection of notes for ES3C5 Signal Processing that I have found useful to have on hand and easily accessible.

The notes made by Adam (MO) cover everything so this is just intended to be an easy to search document.

Use ./generateTables.sh ../src/es2c5/brief-notes.md  in the scripts folder.

3 - Poles and Zeros
General Transfer Function as 2 polynomials
Factorised Transfer Function
Real system as real
Zero DefinitionRoots z of the numerator. When any ,
Pole DefinitionPoles p of the denominator. When any , approaches
Transfer Function GainK is the overall transfer function gain. (Coefficient of and is 1.)
Stable SystemA system is considered stable if its impulse response tends to zero or a finite ...
Components to ResponseReal Components Exponential Response Imaginary angular f...
8 - Signal Conversion between Analog and Digital
Digital Signal Processing WorkflowSee diagram:
SamplingConvert signal from continuous-time to discrete-time. Record amplitude of the an...
OversampleSample too often, use more complexity, wasting energy
UndersampleNot sampling often enough, get
AliasingMultiple signals of different frequencies yield the same data when sampled.
Nyquist Rate
QuantisationThe mapping of
Data InterpolationConvert digital signal back to analogue domain, reconstruct continous signal fro...
Hold CircuitSimplest interpolation in a DAC, where amplitude of continuous-time signal match...
Resolution
Dynamic range
9 - Z-Transforms and LSI Systems
LSI RulesLinear Shift-Invariant
Common Components of LSI SystemsFor digital systems, only need 3 types of LSI circuit components.
Discrete Time Impulse FunctionImpulse response is very similar in digital domain, as it is the system output w...
Impulse Response Sequence
LSI Output
Z-Transform
Z-Transform ExamplesSimple examples...
Binomial Theorem for Inverse Z-Transform
Z-Transform PropertiesLinearity, Time Shifting and Convolution
Z-Transform of Output Signal
Finding time-domain output of an LSI SystemTransform, product, inverse.
Difference EquationTime domain output directly as a function of time-domain input as ...
Z-Transform TableSee table...
12 - Filter Difference equations and Impulse responses
Z-Domain Transfer Function
General Difference Equation
Example 12.1 Proof y[n] can be obtained directly from H[z]See image...
Order of a filter
Taps in a filterMinimum number of unit delay blocks required. Equal to the order of the filter.
Example 12.2 Filter Order and TapsSee example...
Tabular Method for Difference EquationsGiven a difference equation, and its input x[n], can write specific output y[n] ...
Example 12.3 Tabular Method ExampleSee example
Infinite Impulse Response (IIR) FiltersIIR filters have
Example 12.4 IIR FilterSee example
Finite Impulse Response (FIR) FiltersFIR Filter are none recursive (ie, no feedback components), so a[k] = 0 for k!=0...
FIR Difference Equation
FIR Transfer function
FIR Transfer Function - Roots
FIR StabilityFIR FILTERS ARE ALWAYS STABLE. As in transfer function, all M poles are all on t...
FIR Linear Phase ResponseOften have a linear phase response. The phase shift at the output corresponds to...
FIR Filter ExampleSee example 12.5
Ideal Digital FiltersFour main types of filter magnitude responses (defined over 0 \le \Omega \le \p...
Realising Ideal Digital FiltersUse poles and zeros to create simple filters. Only need to consider response ove...
Example 12.6 - Simple High Pass Filter DesignSee diagram
13 - FIR Digital Filter Design
Realising Ideal Digital FilterAim is to get as close as possible to
Practical Digital FiltersGood digital low pass filter will try to realise the (unrealisable) ideal respon...
Windowing Criteria
Practical FIR Filter Design Example 13.2See example...
Specification for FIR Filters Example 13.3See example...
14 - Discrete Fourier Transform and FFT
Discrete Fourier Transform DFT
Inverse DFTx[n] = \frac{1}{N}\sum_{k=0}^{N-1}X[k]e^{jnk\frac{2\pi}{N}}, \quad n=\left { 0,1,2, \cdots , N-1 \right }
Example 14.1 DFT of SinusoidSee example
Zero PaddingArtificially increase the length of the time domain signal by adding zero...
Example 14.2 Effect of Zero PaddingSee example
Fast Fourier Transform FFTFamily of alogrithms that evaluate DFT with complexity of compare...
16 - Digital vs Analogue Recap
Aperiodic (simple periodic) continuous-time signal f(t)Laplace, fourier transform.
More Complex Continuous-time signal f(t)Fourier series, multiples of fundamental, samples of frequency response.
Discrete-time signal f[n] (infinite length)Z-Domain, Discrete-time fourier transform
Discrete-time signal f[n] (finite length)Finite Length N, convert to frequency domain (DFT), N points distributed over 2 ...
StabilityS-domain: negative real component, Z domain: poles within unit circle.
Bi-LinearityNot core module content.
17 - Probabilities and random signals
Random VariableA quantity that takes a non-deterministic values (ie we don't know what the valu...
Probability DistributionDefines the probability that a random variable will take some value.
Probability Density Function (PDF) - Continuous random variables
Probability mass function (PMF) - Discrete random variables
Moments
Uniform DistributionEqual probability for a random variable to take any value in its domain, ie over...
BernoulliDiscrete probability distribution with only 2 possible values (yes no, 1 0, etc)...
Gaussian (Normal) DistributionContinuous probability distribution over , where values closer...
Central Limit Theorem (CLT)Sum of independent random variables can be approximated with Gaussian distributi...
Independent Random VariablesNo dependency on each other (i.e., if knowing the value of one random variable g...
Empirical DistributionsScaled histogram by total number of samples.
Random SignalsRandom variables can appear in signals in different ways, eg:
18 - Signal estimation
Signal EstimationSignal estimation, refers to estimating the values of parameters embedded in a s...
Linear ModelSee equation
Generalised Linear FromSee equation
Optimal estimateSee equation
Predicted estimateSee equation
Observation Matrix See below
Mean Square Error (MSE)See equation
Example 18.1See example
Example 18.2See example
Linear Regression
Weighted Least Squares EstimateWeighted least squares, includes a weight matrix W, where each sample associated...
Maximum Likelihood Estimation (MLE)See equation
19 - Correlation and Power spectral density
CorrelationCorrelation gives a measure of time-domain similarity between two signals.
Cross Correlation
Example 19.1 - Discrete Cross-CorrelationSee example
AutocorrelationCorrelation of a signal with itself, ie or
Example 19.2 - Discrete AutocorrelationSee example
Example 19.3 - Correlation in MATLABSee example
20 - Image Processing
Types of colour encodingBinary (0, 1), Indexed (colour map), Greyscale (range 0->1), True Colour (RGB)
NotationSee below
Digital Convolution
Example 20.1 - 1D Discrete ConvolutionSee example
Example 20.2 - Visual 1D Discrete ConvolutionSee example
Image FilteringDetermine output y[i][j] from input x[i][j] through filter (kernel) h[i][j]
KernelsDifferent types of kernels.
Example 20.3 - Image FilteringSee example

# Part 1 - Analogue Signals and Systems

## Laplace Conversion

### Laplace Table

Insert table here

### Finding Time Domain Output

1. Transform and into Laplace domain
2. Find product
3. Take inverse Laplace transform

Then , so .

Then , so .

### LTI System Properties

LTI = Linear Time Invariant.

• LTI systems are linear. Given system and signals , etc
• LTI is scalable (or homogeneous)
• LTI is time-invariant, ie, if output then:

## 3 - Poles and Zeros

### Factorised Transfer Function

Is factorised and rewrite as a ratio of products:

### Real system as real

Where the numerator i a th order polynomial with coefficients s and the denominator is a th order polynomial with coefficients s. For a system to be real, the order of the numerator polynomial must be no greater than the order of the denominator polynomial, ie: .

### Zero Definition

Roots z of the numerator. When any ,

### Pole Definition

Poles p of the denominator. When any , approaches

### Transfer Function Gain

K is the overall transfer function gain. (Coefficient of and is 1.)

### Stable System

A system is considered stable if its impulse response tends to zero or a finite value in the time domain.

Requires all real components to be negative (on the left hand side of the complex s-plane of a pole-zero plot (left if the imaginary s axis)).

### Components to Response

Real Components Exponential Response Imaginary angular frequency of oscillating responses.

## 4 - Analog Frequency Response

### Frequency Response

Frequency response of a system = output in response to sinusoid input of unit magnitude and specified frequency, . Response is measured as magnitude and phase angle.

### Continuous Fourier Transform

Laplace transform evaluated on the imaginary s-axis at some frequency .

### Magnitude of Frequency Response (MFR)

In words, the magnitude of the frequency response (MFR) is equal to the gain multiplied by the magnitudes of the vectors corresponding to the zeros, divided by the magnitudes of the vectors corresponding to the poles.

### Phase Angle of Frequency Response (PAFR) -

In words, the phase angle of the frequency response (PAFR) is equal to the sum of the phases of the vectors corresponding to the zeros, minus the sum of the phases of the vectors correspond to the poles, plus if the gain is negative.

Each phase vector is measured from the positive real s-axis (or a line parallel to the real s-axis if the pole or zero is not on the real s-axis).

## 5 - Analog Filter Design

### Ideal Filters

Each ideal filter has unambiguous pass bands, which are ranges of frequencies that pass through the system without distortion, and stop bands, which are ranges of frequencies that are rejected and do not pass through the system without significant loss of signal strength. The transition band between stop and pass bands in ideal filters has a size of 0; transitions occur at single frequencies.

### Realisability

System starts to respond to input before input is applied. Non-zero for .

### Causality

Output depends only on past and current inputs, not future inputs.

### Realising Filters

Realise as we seek smooth behaviour.

• Drop for ()
• Would not get suitable behaviour in frequency domain, as discarded 50% of system energy
• But can tolerate delays
• So shift sinc to the right
• Time domain shift = scaling by complex exponential in laplace
• True in fourier transform, so delay in time maintains magnitude but changes phase of frequency response
• Truncate
• As can't wait for infinity, so truncate impulse response.

### Transfer Function of Nth Order Butterworth Low Pass Filter

Butterworth = Maximally flat in pass band (freq response magnitudes are flat as possible for given order)

• = nth pole
• =
• =
• Form semi-circle to left of imaginary s-axis
• = half-power cut-off frequency
• Frequency where filter gain is or

### Frequency Response of common Low pass Butterworth filter

Increasing order improves approximation of ideal behaviour

### Normalised Frequency Response of common Low pass Butterworth filter

To convert normalised frequency form to non-normalised = multiply by the actual

### Minimum Order for Low Pass Butterworth

Round up as want to over-satisfy not under-satisfy

Gain in dB

Gain in dB

## 6 - Periodic Analogue Functions

### Fourier Series

Period signal = sum of complex exponentials.

Fundamental frequency , such that all frequencies in signal are multiples of .

Fundamental period

Fourier spectra only exist at harmonic frequencies (ie integer multiples of fundamental frequency)

### Fourier Coefficients

Important property of Fourier series is how is represents real signals .

• Even magnitude spectrum
• Odd phase spectrum =

Where

### Output of LTI system from Signal with multiple frequency components

Or in other words:

The output of an LTI system due to a signal with multiple frequency components can be found by superposition of the outputs due to the individual frequency components. IE system will change amplitude and phase of each frequency in the input.

### Filtering Periodic Signal (Example 6.2)

See example 6.2 below...

## 7 - Computing with Analogue Signals

This topic isn't examined as it is MATLAB

## 8 - Signal Conversion between Analog and Digital

### Digital Signal Processing Workflow

See diagram:

• Low pass filter applied to time-domain input signal to limit frequencies
• An analogue-to-digital converter (ADC) samples and quantises the continuous time analogue signal to convert it to discrete time digital signal .
• Digital signal processing (DSP) performs operations required and generates output signal .
• A digital-to-analogue converter (DAC) uses hold operations to reconstruct an analogue signal from
• An output low pass filter removes high frequency components introduced by the DAC operation to give the final output .

### Sampling

Convert signal from continuous-time to discrete-time. Record amplitude of the analogue signal at specified times. Usually sampling period is fixed.

### Oversample

Sample too often, use more complexity, wasting energy

### Undersample

Not sampling often enough, get aliasing of our signal (multiple signals of different frequencies yield the same data when sampled.)

### Aliasing

Multiple signals of different frequencies yield the same data when sampled.

If we sample the black sinusoid at the times indicated with the blue marker, it could be mistaken for the red dashed sinusoid. This happens when under-sampling, and the lower signal is called the alias. The alias makes it impossible to recover the original data.

### Nyquist Rate

Minimum ant-aliasing sampling Frequency.

Frequencies above this remain distinguishable.

### Quantisation

The mapping of continuous amplitude levels to a binary representation.

IE: bits then there are quantisation levels. ADC Word length .

Continuous amplitude levels are approximated to the nearest level (rounding). Resulting error between nearest level and actual level = quantisation noise

### Data Interpolation

Convert digital signal back to analogue domain, reconstruct continous signal from discrete time series of points.

### Hold Circuit

Simplest interpolation in a DAC, where amplitude of continuous-time signal matches that of the previous discrete time signal.

IE: Hold amplitude until the next discrete time value. Produces staircase like output.

### Resolution

Space between levels, often represented as a percentage.

For -bit DAC, with uniform levels

### Dynamic range

Range of signal amplitudes that a DAC can resolve between its smallest and largest (undistorted) values.

## 9 - Z-Transforms and LSI Systems

### LSI Rules

Linear Shift-Invariant

### Common Components of LSI Systems

For digital systems, only need 3 types of LSI circuit components.

1. A multiplier scales the current input by a constant, i.e., .
2. An adder outputs the sum of two or more inputs, e.g., .
3. A unit delay imposes a delay of one sample on the input, i.e, .

### Discrete Time Impulse Function

Impulse response is very similar in digital domain, as it is the system output when the input is an impulse.

### LSI Output

Discrete Convolution of input signal with the impulse response.

### Z-Transform

Converts discrete-time domain function into complex domain function , in the z-domain Assume is causal, ie

Discrete time equivalent to Laplace Transform. However can be written by direct inspection (as have summation instead of intergral). Inverse equally as simple.

### Z-Transform Examples

Simple examples...

### Binomial Theorem for Inverse Z-Transform

Cannot always find inverse Z-tranform by immediate inspection, in particular if the Z-transform is written as a ratio of polynomials of z. Can use Binomial theorem to convert into single (sometimes infinite length) polynomial of

### Z-Transform Properties

Linearity, Time Shifting and Convolution

See example

### Z-Transform of Output Signal

Where = Pulse Transfer Function (as it is also the system output when the time-domain input is a unit impulse.) but by convention can refer to as the Transfer Function

### Finding time-domain output of an LSI System

Transform, product, inverse.

1. Transform and into z-domain
2. Find product
3. Taking the inverse Z-transform of

### Difference Equation

Time domain output directly as a function of time-domain input as well as previous time-domain outputs (ie can be feedback).

See table...

## 10 - Stability of Digital Systems

### Z-Domain Transfer Function

Negative powers of z.

No constraint on and to be real (unlike analogue) but often assume

### Poles and Zeros of Transfer Function

• Coefficient of each in this form is 1.
• Poles and zeros carry same meaning as analogue
• Unfortunately symbol for variable and zeros are very similar (take care)
• Insightful to plot

### Bounded Input and Bounded Output (BIBO) Stability

Stable if bounded input sequence yields bounded output sequence.

A system is BIBO stable if all of the poles lie inside the unit circle

A system is Conditionally stable if there is atleast 1 pole directly on the unit circle.

Explanation:

• An input sequence is bounded if each element in the sequence is smaller than some value .
• An output sequence corresponding to is bounded if each element in the sequence is smaller than some value .

## 11 - Digital Frequency Response

### LSI Frequency Response

Output in response to a sinusoid input of unit magnitude and some specified frequency. Shown in two plots (magnitude and phase) as a function of input frequency.

### Discrete-Time Fourier Transform (DTFT) - Digital Frequency Response

Where angle is the angle of th unit vector measured from the positive real -axis. Denotes digital radial frequency, measured in radians per sample

as spectrum of (frequency response).

Convention of writing DTFT includes or simply

Derivation: Using Z-Transform Definition.

Let be polar coords (), ie magnitude to r, angle . Hence rewrite

Then let , so that any point lies on the unit circle.

### LSI Transfer Function

is a function of vectors from the system's poles and zeros to the unit circle at angle .Thus from pole-zero plot, can geometrically determine magnitude and phase of frequency response.

### Magnitude of Frequency Response (MFR)

In words, the magnitude of the frequency response (MFR) is equal to the gain multiplied by the magnitudes of the vectors corresponding to the zeros, divided by the magnitudes of the vectors corresponding to the poles.

Repeats every as Eulers Formula. Due to symettery of poles and zeros about real -axis, frequency response is symmetric about , so only need to find over one interval of

### Phase Angle of Frequency Response (PAFR) -

In words, the phase angle of the frequency response (PAFR) is equal to the sum of the phases of the vectors corresponding to the zeros, minus the sum of the phases of the vectors correspond to the poles, plus if the gain is negative.

Repeats every as Eulers Formula. Due to symettery of poles and zeros about real -axis, frequency response is symmetric about , so only need to find over one interval of

See image...

## 12 - Filter Difference equations and Impulse responses

### General Difference Equation

Real coefficients and are the same. (Note = 1, so no coefficient corresponding to ).

Easier to convert directly between transfer function (with negative powers of z) and the difference equation for output , ideal for implementation of the system. (rather than use time-domain impulse response )

See image...

### Taps in a filter

Minimum number of unit delay blocks required. Equal to the order of the filter.

See example...

### Tabular Method for Difference Equations

Given a difference equation, and its input x[n], can write specific output y[n] using tabular method.

1. Starting with input , make a column for every input and output that appears in difference equation
2. ASsume every output and delayed input is initially zero (ie the filter is causal, initially no memory, hence system is quiescent)
3. Fill in column for with given system input for all rows needed, and fill in delayed versions of
4. Evaluate from inital input, and propagate the value of y[0] to delayed outputs (as relavent)
5. Evaluate from s and
6. Continue evaluating output and propagating delayed outputs.

Can be alternative method for finding time-domain impulse response

See example

### Infinite Impulse Response (IIR) Filters

IIR filters have infinite length impulse responses because they are recursive (ie feedback terms associated with non-zero poles in transfer function, hence exists.)

Standard transfer function and difference equation can be used to represent.

Not possible to have a linear phase response (so there are different delays associated with different frequencies, and they are not always stable (depending on the exact locations of poles.))

IIR filters are more efficient than FIR designs at controlling gain of response.

Although response is technically infinite, in practice decays towards zero or can be truncated to zero (assume response is beyond some value )

See example

### Finite Impulse Response (FIR) Filters

FIR Filter are none recursive (ie, no feedback components), so a[k] = 0 for k!=0.

Finite in length, and strictly zero beyond that ( for ). Therefore the number of filter taps dictates the length of an FIR impulse response

Since there is no feedback, can write impulse response as:

### FIR Transfer function

Simplified from general differernce equation tranfer function.

### FIR Transfer Function - Roots

More convenient to work with positive powers of z, so multiply top and bottom by then factor.

### FIR Stability

FIR FILTERS ARE ALWAYS STABLE. As in transfer function, all M poles are all on the origin (z =0) and so always in the unit circle.

### FIR Linear Phase Response

Often have a linear phase response. The phase shift at the output corresponds to a time delay.

See example 12.5

### Ideal Digital Filters

Four main types of filter magnitude responses (defined over , mirrored over and repeated every )

• Low Pass - pass frequencies less than cut-off frequency and reject frequencies greater.
• High Pass - rejects frequencies less than cut-off frequency and pass frequencies greater.
• Band Pass - Passes frequency within specified range, ie between and , and reject frequencies that are either below or above the band within
• Band Stop - Rejects frequency within specified range, ie between and , and passes all other frequencies within

Ideal response appear to be fundamentally different from ideal analogue, however we only care over fundamental band where behaviour is identical

### Realising Ideal Digital Filters

Use poles and zeros to create simple filters. Only need to consider response over the fixed band.

Key Concepts:

• To be physically realisable, complex poles and zeros need to be in conjugate pairs

• Can place zeros anywhere, so will often place directly on unit circle when frequency / range of frequency needs to be attenuated

• Poles on the unit circle should generally be avoided (conditionally stable). Can try to keep all poles at origin so can be FIR, otherwise IIR, so feedback. Poles used to amplify response in the neighbourhood of some frequency.

• Low Pass - zeros at or near , poles near which can amplify maximum gain, or be used at a higher frequency to increase size of pass band.

• High Pass - literally inverse of low pass. Zeros at or near , poles near which can amplify maximum gain, or be used at a lower frequency to increase size of pass band.

• Band Pass - Place zeros at or near both and ; so must be atleast second order. Place pole if needed to amplify the signal in the neighbourood of the pass band.

• Band Stop - Place zeros at or near the stop band. Zeros must be complex so such a filter must be atelast second order. Place poles at or near both and if needed.

See diagram

## 13 - FIR Digital Filter Design

As long as - otherwise there will be an alias at a lower frequency. So to avoid aliasing.

### Realising Ideal Digital Filter

Aim is to get as close as possible to ideal behaviour. But when using Inverse DTFT, the ideal impulse response is .

This is analogous to the inverse Fourier transform of the ideal analogue low pass frequency response.

Sampled a scaled sinc function, non-zero values for . So needs to respond to an input before the input is applied, thus unrealisable.

### Practical Digital Filters

Good digital low pass filter will try to realise the (unrealisable) ideal response. Will try to do this with FIR filters (always stable, tend to have greater flexibility to implement different frequency responses).

• Need to induce a delay to capture most of the ideal signal energy in causal time, ie: use
• Truncate response to delay tolerance , such that for . Also limits complexity of filter: shorter = smaller order
• Window response, scales each sample, attempt to mitigate negative effects of truncation

### Windowing

Window Method - design process: start with ideal and windowing infinite time-domain response to obtain a realsiable that can be implemented.

### Windowing Criteria

• Main Lobe Width - Width in frequency of the main lobe.
• Roll-off rate - how sharply main lobe decreases, measured in dB/dec (db per decade).
• Peak side lobe level - Peak magnitude of the largest side lobe relative to the main lobe, measured in dB.
• Pass Band ripple - The amount the gain over the pass band can vary about unity and
• Pass Band Ripple Parameter, dB-
• Stop Band ripple - Gain over the stop band, must be less than the stop band ripple
• Transition Band -

See example...

See example...

## 14 - Discrete Fourier Transform and FFT

### Discrete Fourier Transform DFT

For

This is Forward discrete Fourier transform. (Not discrete time transform, but samples of it over interval

Explanation:

Discrete-time Fourier Transfomr (DTFT), takes discrete time signal, provides continous spectrem that repeats every . Defined for infinite length sequency , gives continous spectrum with values at all frequencies.

Digital often has finite length sequences. (Also inverse DTFT, uses intergration thus approximated). So assume sequence is length .

Sample spectrum . Repeats every , can sample over .

Take same number of samples in frequency domain as length of time domain signal.So evenly spaced samples of . (Aka bins)

Occur at fundemental frequency

Substitude into the DTFT.

For

### Example 14.1 DFT of Sinusoid

See example

Artificially increase the length of the time domain signal by adding zeros to the end to see more detail in the DTFT as DFT provides sampled view of DTFT, only see DTFT at frequencies.

See example

### Fast Fourier Transform FFT

Family of alogrithms that evaluate DFT with complexity of compared to . Achieved with no approximations.

Details are beyond module, but can be used in matlab with fft function.

## 15 - Computing Digital Signals

This topic isn't examined as it is MATLAB

## 16 - Digital vs Analogue Recap

### Aperiodic (simple periodic) continuous-time signal f(t)

Laplace, fourier transform.

• APeriodic (simple periodic) continuous time signal
• Convert to Laplace domain (s domain) via Laplace transform
• Which is the (continuous) Fourier transform.
• Fourier transform of the signal is its frequency response , generally defined for all
• Laplace and fourier transform, have corresponding inverse transforms, convert or back to

### More Complex Continuous-time signal f(t)

Fourier series, multiples of fundamental, samples of frequency response.

• For a more complex periodic continuous-time signal f (t)
• Fourier series representation decomposes the signal into its frequency components at multiples of the fundamental frequency .
• Can be interpreted as samples of the frequency response ,
• which then corresponds to periodicity of over time.
• The coefficients are found over one period of .

### Discrete-time signal f[n] (infinite length)

Z-Domain, Discrete-time fourier transform

• Discrete-time signal , we can convert to the z-domain via the Z-transform,
• Which for is the discrete-time Fourier transform DTFT.
• The discrete-time Fourier transform of the signal is its frequency response
• Repeats every (i.e., sampling in time corresponds to periodicity in frequency).
• There are corresponding inverse transforms to convert or back to

### Discrete-time signal f[n] (finite length)

Finite Length N, convert to frequency domain (DFT), N points distributed over 2 pi (periodic)

• For discrete-time signal with finite length (or truncated to) N,
• Can convert to the frequency domain using the discrete Fourier transform,
• which is also discrete in frequency.
• The discrete Fourier transform also has N points distributed over and is otherwise periodic.
• Here, we see sampling in both frequency and time, corresponding to periodicity in the other domain (that we usually ignore in analysis and design because we define both the time domain signal and frequency domain signal over one period of length N).

### Stability

S-domain: negative real component, Z domain: poles within unit circle.

### Bi-Linearity

Not core module content.

## 17 - Probabilities and random signals

### Random Variable

A quantity that takes a non-deterministic values (ie we don't know what the value will be in advance).

### Probability Distribution

Defines the probability that a random variable will take some value.

### Probability Density Function (PDF) - Continuous random variables

For a random variable , take values between and (could be ), is the probability that .

The integration of these probabilities is equal to 1.

Can take integral over subset to calculate the probability of X being within that subset.

### Probability mass function (PMF) - Discrete random variables

For a random variable , take values between and (could be ), is the probability that .

The sum of these probabilities is equal to 1.

Can take summation over subset to calculate the probability of X being within that subset.

### Moments

Of PMF and PDF respectively.

• , called the mean - Expected (average) value
• , called the mean-squared value, describes spread of random variable.

Often refer second order moments to as variance . mean-squared value, with correction for the mean

Standard deviation

### Uniform Distribution

Equal probability for a random variable to take any value in its domain, ie over .

PDF continuous version:

Discrete uniform distributions: result of dice roll, coin toss etc. Averege is average of min and max.

### Bernoulli

Discrete probability distribution with only 2 possible values (yes no, 1 0, etc). Values have different probabilities, in general .

Mean: , Variance:

### Gaussian (Normal) Distribution

Continuous probability distribution over , where values closer to mean are more likely.

Arguably most important continuos distribution as appears everywhere.

PDF is

### Central Limit Theorem (CLT)

Sum of independent random variables can be approximated with Gaussian distribution. Approximation improves with as more random variables are included in the sum. True for any probability distributions.

### Independent Random Variables

No dependency on each other (i.e., if knowing the value of one random variable gives you no information to be able to better guess another random variable, then those random variables are independent of each other).

### Empirical Distributions

Scaled histogram by total number of samples.

To observe behaviour that would match a PDF or PMF, require infinite number of samples. In practice can make histogram.

### Random Signals

Random variables can appear in signals in different ways, eg:

• Thermal noise - in all electronics, from agitation electrons. Often modelled by adding gaussian random variable to signal
• Signal processing techniques introduce noise - aliasing, quantisation, non-ideal filters.
• Random variables can be used to store information, e.g., data can be encoded into bits and delivered across a communication channel. A receiver does not know the information in advance and can treat each bit as a Bernoulli random variable that it needs to estimate.
• Signals can be drastically transformed by the world (wireless signals obstructed by buildings trees etc) - Analogue signals passing through unknown system , which can vary with time etc

## 18 - Signal estimation

### Signal Estimation

Signal estimation, refers to estimating the values of parameters embedded in a signal. Signals have noise, so can't just calculate parameters.

### Linear Model

See equation

Polynomial terms,, linearity means y[n] must be linear function of unknown parameters.

EG:

• A,B are unknown parameters
• refers to noise - assume gaussian random variables with mean and varience - also assume white noise.

Write as column vector for each n.

Create observation matrix .

Since there are 2 parameters, matrix.

Can therefore be written as

With Optimal estimate :

Can write prediction:

Calculate MSE

### Generalised Linear From

See equation

Where is a vector of known samples. Convenient when our signal is contaminated by some large interference with known characteristics.

To account for this in the estimator but subtracting by s from both sides.

See equation

See equation

Without noise.

### Observation Matrix

See below

matrix where is the number of parameters, and is the number of time steps.

Each column is the coefficients of the corresponding parameter at the given timestamp (per row).

See equation

See example

See example

### Linear Regression

AKA Ordinary least squares (OLS).

Form of observation matrix had to be assumed but may be unknown. If so can try different ones and find simplest that has best MSE.

### Weighted Least Squares Estimate

Weighted least squares, includes a weight matrix W, where each sample associated with positive weight.

Places more emphasis on more reliable samples.

Good choice of weight :

Therefore resulting in:

Using equation, where W is the column vector of weights.

theta = lscov(Obs, y,W);


### Maximum Likelihood Estimation (MLE)

See equation

Found by determining , maximises the PDF of the signal , which depends on the statistics of the noise .

Given some type of probability distribution, the MLE can be found.

MATLAB mle function from the Statistics and Machine Learning Toolbox.

## 19 - Correlation and Power spectral density

### Correlation

Correlation gives a measure of time-domain similarity between two signals.

### Cross Correlation

• is the time shift of sequence relative to the sequence.
• Approximation as signal lengths are finite and the signals could be random.

See example

### Autocorrelation

Correlation of a signal with itself, ie or

• Gives a measure of whether the current value of the signal says anything about a future value. Especially good for random signals.

Key Properties

1. Autocorrelation for zero delay is the same as the signals mean square value. The auto correlation is never bigger for any non-zero delay.
2. Auto correlation is an even function of or , ie
3. Autocorrelation of sum of two uncorrelated signals is the same as the sums fo the autocorrelations of the two individual signals.

For and are uncorrelated,

See example

See example

## 20 - Image Processing

### Types of colour encoding

Binary (0, 1), Indexed (colour map), Greyscale (range 0->1), True Colour (RGB)

• Binary has value 0 and 1 to represent black and white
• Indexed each pixel has one value corresponding to pre-determined list of colours (colour map)
• Greyscale - each pixel has value within 0 (black) and 1 (white) - often write as whole numbers and then normalise
• True colour - Three associated values, RGB

But focus on binary and greyscale for hand calculations

### Notation

See below

- follows same indexing conventions as MATLAB

ie: refers to vertical coordinate (row)

refers to horizontal coordinate (column)

is the top left pixel.

See example

See example

### Image Filtering

Determine output y[i][j] from input x[i][j] through filter (kernel) h[i][j]

Filter (Kernel) = , assume square matrix with odd rows and columns so obvious middle

1. Flip impulse response to get
1. Achieved by mirroring all elements around center element.
2. By symmetry, sometimes
2. Move flipped impulse response along input image .
3. Each time kernel is moved, multiply all elements of by corresponding covered pixels in .
1. Add together products and store sum in output - corresponds to middle pixel covered by kernel
2. Only consider overlaps between and where the middle element of the kernel covers a pixel in the input image

### Edge Handling

• Zero Padding Treat all off image pixels as having value 0. beyond defined image. Simplest but may lead to unusual artefacts at the edges of the image. Only option available for conv2 and default for imfilter.
• Replicating the border - Assuming off image have same values as nearest element along edge of image. IE: Assume pixels at the outside corner take the value of the corner pixel

### Kernels

Different types of kernels.

Larger kernels have increased sensitivity but more expensive to compute.

• Values add to 0 = Removes signal strength to accentuate certain details

• Values add to 1 = maintain signal strength by redistributing

• Low Pass Filter - Equivalent to taking weight average around neighbourhood of pixel. Adds to 1

• Blurring filter - Similar to low pass, but elements adds uo to more than 1, so washes out the image more

• High Pass Filter - Accentuates transitions between colours, can be used as simple edge detection (important task, first step to detecting objects)

• Sobel operator - More effective at detecting edges than high pass filter. Do need to apply different kernels for different directions. X-gradient = detecting vertical edges, Y-gradient = detecting horizontal edges

See example