Dynamiske systemer oplæsning PDF

Indhold {#indhold.Overskrift} ======= [Sensor technology 3](#sensor-technology) [Lecture 1 -- Basics of sensors 3](#lecture-1-basics-of-sensors) [Lecture 3 -- Resistive sensors 3](#lecture-3-resistive-sensors) [Lecture 4 -- Capacitive and inductive sensors 4](#lecture-4-capacitive-and-inductive-...

Indhold {#indhold.Overskrift} ======= [Sensor technology 3](#sensor-technology) [Lecture 1 -- Basics of sensors 3](#lecture-1-basics-of-sensors) [Lecture 3 -- Resistive sensors 3](#lecture-3-resistive-sensors) [Lecture 4 -- Capacitive and inductive sensors 4](#lecture-4-capacitive-and-inductive-sensors) [Lecture 5 -- Microphone 4](#lecture-5-microphone) [Signal analysing 5](#signal-analysing) [Lecture 1 -- Signals and statistics 5](#lecture-1-signals-and-statistics) [Lecture 2 -- Number representation 5](#lecture-2-number-representation) [Lecture 3 -- Linear systems 5](#lecture-3-linear-systems) [Lecture 3 -- Fourier transform 5](#lecture-3-fourier-transform) [Lecture 4 -- Frequency plot 6](#lecture-4-frequency-plot) [Lecture 5 -- Convolution 6](#lecture-5-convolution) [Lecture 6 -- Heart sounds 6](#lecture-6-heart-sounds) [Lecture 7 -- filters 6](#lecture-7-filters) [Lecture 7 -- moving average 7](#lecture-7-moving-average) [Lecture 8 -- FIR Frequency separation filters 7](#lecture-8-fir-frequency-separation-filters) [Lecture 9 -- IIR filters 7](#lecture-9-iir-filters) [Programming 9](#programming) [Lecture 1 9](#lecture-1) [Lecture 2 10](#lecture-2) [Lecture 3 10](#lecture-3) [Lecture 4 11](#lecture-4) [Lecture 5 11](#lecture-5) [Lecture 6 12](#lecture-6) [Questions that may be good to be able to answer before final exam 12](#questions-that-may-be-good-to-be-able-to-answer-before-final-exam) [Explanation of MATLAB code from project 14](#explanation-of-matlab-code-from-project) [Step 1 -- Loading of the data 14](#step-1-loading-of-the-data) [Step 2 -- Editing the time data 15](#step-2-editing-the-time-data) [Step 3 -- Data validations 16](#step-3-data-validations) [Step 4 -- Calculating the sampling frequency 17](#step-4-calculating-the-sampling-frequency) [**Step 5 -- Removal of the DC-offset** 18](#step-5-removal-of-the-dc-offset) [Step 6 - FFT (Fast fourier transformation) 19](#step-6---fft-fast-fourier-transformation) [Step 7 -- Filter design and application 21](#step-7-filter-design-and-application) [Step 8 -- FFT after filtering 22](#step-8-fft-after-filtering) [Step 9A -- Identification of the peaks 24](#step-9a-identification-of-the-peaks) [Step 9B -- Highlighting the peaks of S2 25](#step-9b-highlighting-the-peaks-of-s2) [Step 10 -- Calculation of the heart rate 27](#step-10-calculation-of-the-heart-rate) [Explanation of Code Structure and Purpose of Key Functions and Decisions 29](#explanation-of-code-structure-and-purpose-of-key-functions-and-decisions) [Forsøg 1A - Amplitude Change 29](#fors%C3%B8g-1a---amplitude-change) [Forsøg 1B - Frequency Change 29](#fors%C3%B8g-1b---frequency-change) [Databehandling - Data Plotting and Fitting 30](#databehandling---data-plotting-and-fitting) [\ ] [Sensor technology] =============================== +-----------------------------------+-----------------------------------+ | Lecture 1 -- Basics of sensors | | | ------------------------------ | | +===================================+===================================+ | Passive vs Active sensor | | +-----------------------------------+-----------------------------------+ | Transfer function | What is a transfer function? | +-----------------------------------+-----------------------------------+ | | Why is the inverse of a transfer | | | function often interesting? | +-----------------------------------+-----------------------------------+ | | A transfer function | | | | | | - Can have many functions | | | | | | - We often linearize it | +-----------------------------------+-----------------------------------+ | Three types of Calibration | | +-----------------------------------+-----------------------------------+ | | Grinding | +-----------------------------------+-----------------------------------+ | | Trimming | +-----------------------------------+-----------------------------------+ | | Calculation of the transfer | | | function | +-----------------------------------+-----------------------------------+ | Terms of sensor specification | | +-----------------------------------+-----------------------------------+ | | Sensitivity | +-----------------------------------+-----------------------------------+ | | Hysteresis | +-----------------------------------+-----------------------------------+ | | Saturation | +-----------------------------------+-----------------------------------+ | | Resolution | +-----------------------------------+-----------------------------------+ | Lecture 3 -- Resistive sensors | | | ------------------------------ | | +-----------------------------------+-----------------------------------+ | What is a resistive sensor | | +-----------------------------------+-----------------------------------+ | What parameters define the | [I, A, T ]{.smallcaps} | | resistance of a cable | | +-----------------------------------+-----------------------------------+ | Piezoelectric effect | | +-----------------------------------+-----------------------------------+ | Strain Sensor | How does it work? | +-----------------------------------+-----------------------------------+ | Force sensor | How does it work? | +-----------------------------------+-----------------------------------+ | | FSR | +-----------------------------------+-----------------------------------+ | | FSC | +-----------------------------------+-----------------------------------+ | Temperature sensor | How does it work? | +-----------------------------------+-----------------------------------+ | | Thermistor(Resistive type) | +-----------------------------------+-----------------------------------+ | | Pyroelectric | +-----------------------------------+-----------------------------------+ | | Thermocouples | +-----------------------------------+-----------------------------------+ | How to measure change in | Wheatstone bridge | | resistance | | | | - How does it look like? | | | | | | - What are the advantages in | | | comparison with a voltage | | | divider? | +-----------------------------------+-----------------------------------+ | How to amplify small voltages | Intrusmental amplifier | | | | | | - Properties: High input | | | resistanc, high CMRR | +-----------------------------------+-----------------------------------+ | | | +-----------------------------------+-----------------------------------+ | Lecture 4 -- Capacitive and induc | | | tive sensors | | | --------------------------------- | | | ------------ | | +-----------------------------------+-----------------------------------+ | What is capacitance | How does it work? | +-----------------------------------+-----------------------------------+ | | What parameters change the | | | capacitance of a plate capacitor? | +-----------------------------------+-----------------------------------+ | Humidity sensor | | +-----------------------------------+-----------------------------------+ | Condenser microphone | | +-----------------------------------+-----------------------------------+ | Touch screen | | +-----------------------------------+-----------------------------------+ | Lecture 5 -- Microphone | | | ----------------------- | | +-----------------------------------+-----------------------------------+ | Different types of microphones | Condenser | | | | | | - Principle | | | | | | - Advantages and disadvantages | +-----------------------------------+-----------------------------------+ | | Dynamic | | | | | | - Principle | | | | | | - Advantages and disadvantages | +-----------------------------------+-----------------------------------+ | | Ribbon | | | | | | - Principle | | | | | | - Advantages and disadvantages | +-----------------------------------+-----------------------------------+ | | Carbon | | | | | | - Principle | | | | | | - Advantages and disadvantages | +-----------------------------------+-----------------------------------+ | Frequency response | | +-----------------------------------+-----------------------------------+ | Directionality | | +-----------------------------------+-----------------------------------+ | Principle of MEMS | | +-----------------------------------+-----------------------------------+ | Used for accelerometer | | +-----------------------------------+-----------------------------------+ [Signal analysing] ============================== +-----------------------------------+-----------------------------------+ | Lecture 1 -- Signals and statisti | | | cs | | | --------------------------------- | | | -- | | +===================================+===================================+ | Signal | A description of how one | | | parameter depends on another. | +-----------------------------------+-----------------------------------+ | Continuous signal | A signal that can change | | | infinitely, often representation | | | analog | +-----------------------------------+-----------------------------------+ | Discrete signal | A signal form from parameters | | | that are quantized (from analog | | | to digital converter) are said to | | | de discrete or digitized signal. | +-----------------------------------+-----------------------------------+ | Time domain | A signal that used time as the | | | independent variable (x-axis) is | | | said to be in the time domain. | | | | | | If the x-axis is labelled with | | | something like sample number, | | | then they are in the time domain, | | | as sampling at equal intervals of | | | time is the most common way of | | | obtaining signals. | +-----------------------------------+-----------------------------------+ | Frequency domain | A signal that uses frequency as | | | the independent variables is said | | | to in the frequency domain. | +-----------------------------------+-----------------------------------+ | Analog to digital conversion | How a continues signal (analog) | | | into a discrete signal (digital) | | | | | | - The higher the sample rate | | | the more information you | | | received | | | | | | - The higher the resolution the | | | more changes you can see | +-----------------------------------+-----------------------------------+ | | Happens in two steps | | | | | | 1. Sampling | | | | | | a. Conversion of the | | | independent variables | | | from continues to | | | discrete | | | | | | 2. Quantization | | | | | | b. Conversion of the | | | depended variables from | | | continuous to discrete | +-----------------------------------+-----------------------------------+ | Sampling | Sampling is the process of taking | | | measurement of a continuous | | | signal at regular time intervals | | | to create a discrete version of | | | the signal | | | | | | - Sampling converts an analog | | | signal into a series of | | | discrete values | | | | | | - Each sample represents the | | | signal amplitude at a | | | specific point in time. | | | | | | - The quality of the digitized | | | signal depends on the | | | sampling frequency and the | | | resolution of the ADC | +-----------------------------------+-----------------------------------+ | | Sampling frequency or sampling | | | rate is the number of samples | | | taken per second when a signal is | | | being digitized. It's measured in | | | Hz. | | | | | | \ | | | [\$\$f\_{s} = \\frac{1}{T\_{s}}\\ | | | \$\$]{.math.display}\ | +-----------------------------------+-----------------------------------+ | | Nyquist theorem | | | | | | - To accurately reconstruct a | | | signal, **the sampling | | | frequency must be at least | | | twice the highest frequency | | | present in the analog | | | signal**. | | | | | | - If a signal contains | | | frequencies op to 10 kHz, | | | then the sampling frequency | | | should be at least 20 kHz. | | | | | | - If the sampling rate is too | | | low aliasing occurs | | | | | | - **Aliasing**: misinterpreting | | | frequencies due to | | | insufficient sampling | +-----------------------------------+-----------------------------------+ | Resolution | **Resolution** referees to the | | | level of detail or precision in a | | | digital representation of an | | | analog signal | | | | | | Determined by the number of bits | | | used to represent the signal | | | | | | **More bits = higher resolution = | | | more precise representation** | +-----------------------------------+-----------------------------------+ | | **Most significant bit (MSB)** is | | | the bit in a binary number that | | | represents the largest value. | | | | | | - Contributes the most weight | | | in a binary representation. | | | | | | - In a ADC system, the MSB | | | determines whether the | | | digital value is closer to | | | the maximum possible level | | | | | | - If a ADC has a output of 8 | | | bit number, then the MSB can | | | represent ½ of the full-scale | | | range. | +-----------------------------------+-----------------------------------+ | Quantization | The process of converting a | | | continuous analog signal into a | | | discrete digital signal during | | | analog-to-digital conversion. | | | | | | Process: | | | | | | 1. The continues range of signal | | | values is divided into a | | | finite number of levels | | | | | | 2. Each sample of the analog | | | signal is assigned to the | | | nearest discrete level | | | | | | a. Example: A 3 bit ADC, | | | there are 2\^3 = 8 levels | | | available. | | | | | | b. If a analog signal falls | | | between the two levels, | | | its rounded to the | | | nearest level, which can | | | give means that the | | | digital output is equal | | | to the input plus a | | | quantization error. | +-----------------------------------+-----------------------------------+ | | **Quantization error** is the | | | difference between the original | | | continuous signal and the | | | quantized digital signal. | | | | | | - The maximum error for any | | | sample happens when the | | | analog value is exactly | | | halfway between two adjacent | | | quantication levels. | | | | | | - That is +- ½ least | | | significant bit | | | | | | - The error introduced by | | | approximating continuous | | | values | +-----------------------------------+-----------------------------------+ | | **The LSB** is the smallest step | | | the ADC can represent. It's the | | | distance between two adjacent | | | quantization levels. | | | | | | If an ADC have a 10 bit | | | resolution and the input is 0 to | | | 5 volts | | | | | | - Total amount of levels = | | | amount of bits to a power of | | | 2 = [2^10^ = 1024]{.math | | |.inline} | | | | | | - LSB = | | | [\$\\frac{\\text{Range}}{\\te | | | xt{Number\\ | | | of\\ levels\\ }} = \\frac{5\\ | | | V}{1024\\ } \\approx 4.88\\ | | | m\\ V\\ \$]{.math.inline} | | | | | | - Therefor the maximum | | | quantization error is +- ½ | | | LSB or about +- 2.44 m V | +-----------------------------------+-----------------------------------+ | | **Random noise or quantization | | | error** | | | | | | - Quantization error often | | | resembles random noise, as | | | its uncorrelated with the | | | signal -- e.g. it doesn't | | | follow the pattersn of the | | | input signal | | | | | | - Error varies randomly as the | | | signal fluctutates between | | | the quantiation levels | | | | | | - The erro looks like white | | | noise (uniformly distribued | | | acrous frequencies) when | | | analysed in the frequency | | | domain | | | | | | - The randomness is more true | | | when the signal is complex | | | and the resolution of the ADC | | | is high | | | | | | **Its unavoidable artefact, | | | therefore important to be aware | | | of it.** | | | | | | **It can be minimized by having a | | | higher resolution ADC -\> more | | | bits = smaller LSB** | +-----------------------------------+-----------------------------------+ | Statistics | **Mean** is the average value of | | | a signal. | | | | | | The mean is in electronics | | | commonly called DC (Direct | | | current) value, the AC | | | (Alternating current) referees to | | | how the signal fluctuates around | | | the mean value. | +-----------------------------------+-----------------------------------+ | | **Standard deviation** is the | | | measure of how far the signal | | | fluctuates from the mean. | +-----------------------------------+-----------------------------------+ | | **Variance** represents the power | | | of the standard deviation | +-----------------------------------+-----------------------------------+ | | **RMS (Root-mean-square)** | | | measure both the AC and DC | | | standard deviation, If a signal | | | do not have a DC component, the | | | RMS value is identical to its | | | standard deviation | +-----------------------------------+-----------------------------------+ | | **Signal-to-noise (SNR)** is | | | equal to the mean divided by the | | | standard deviation. | +-----------------------------------+-----------------------------------+ | | **Histogram** displays the number | | | of samples there are in the | | | signal that have each of these | | | possible values. Th sum of all | | | the values in the histogram must | | | be equal to the number of points | | | in the signal. | | | | | | The ***histogram*** is what is | | | formed from an acquired signal. | | | The curve for the process is | | | called the probability mass | | | function (pmf) | | | | | | The **pmf** describes the | | | *probability* that a certain | | | value will be generated. | +-----------------------------------+-----------------------------------+ | | Normal distribution the shape of | | | the usually bell shape | | | probability density function | +-----------------------------------+-----------------------------------+ | | Power spectrum | +-----------------------------------+-----------------------------------+ | | Density | +-----------------------------------+-----------------------------------+ | Precision vs Accuracy | Used to describe system and | | | methods that measure, estimate | | | and predict. | | | | | | Want to know the true value or | | | truth of the measured value. | | | Precision and accuracy are ways | | | of describing the error that can | | | happen. | +-----------------------------------+-----------------------------------+ | | **Precision** is a measure of | | | random noise. | | | | | | How much is the value is | | | fluctuating around. | | | | | | **Can improved by doing many | | | measurements**. | | | | | | To which extent is the individual | | | measurements do not agree with | | | each other. | +-----------------------------------+-----------------------------------+ | | **Accuracy** is a measure of | | | calibration. | | | | | | How close a measure is to the | | | true values. | | | | | | **Can be improved by | | | calibration.** | +-----------------------------------+-----------------------------------+ | Lecture 2 -- Number representatio | | | n | | | --------------------------------- | | | - | | +-----------------------------------+-----------------------------------+ | Integer (Fixed point) | **Unsigned** = Positive only (0 | | | to 255) | | | | | | **Signed** = Including negatives | | | ( -127 to 128) | | | | | | Representation: | | | | | | - **Offset binary**: Shifted | | | range for negatives | | | | | | - **Sign and magnitude**: | | | leftmost bit indicates sign | | | | | | - **Two's complement:** Most | | | commonly used, efficient for | | | hardware operations | +-----------------------------------+-----------------------------------+ | Real numbers (Floating point) | - Similar to scientific | | | notation but used base 2 | | | | | | - Example: | | | [*v* = (−1)^*S*^   *M*   2^(* | | | E*−127)^]{.math | | |.inline} | | | | | | - IEEE standards | | | | | | - **Single** precision (32 | | | bit) | | | [ ∼ 10^ − 38^ *to*  ∼ 10^ | | | 38^]{.math | | |.inline} | | | | | | - **Double** precision (64 | | | bit) | | | [ ∼ 10^ − 308^ *to*  ∼ 10 | | | ^308^]{.math | | |.inline} | | | | | | - MATLAB uses double precision | | | as default. | +-----------------------------------+-----------------------------------+ | Number precision | Integer = Fixed spacing between | | | numbers | +-----------------------------------+-----------------------------------+ | | Floats = Varies with magnitude | | | and lager gaps for larger numbers | | | | | | Limitations of floats | | | | | | - Gaps between representable | | | numbers increase with value | | | | | | - In loops, **rounding errors | | | can accumulate**, leading to | | | skipped values. | +-----------------------------------+-----------------------------------+ | | Round-off error | +-----------------------------------+-----------------------------------+ | Lecture 3 -- Linear systems | | | --------------------------- | | +-----------------------------------+-----------------------------------+ | Definition of linear system | System = Any process that | | | produces an output signal in | | | response to an input signal | +-----------------------------------+-----------------------------------+ | | A system is linear if its | | | satisfies: | | | | | | 1. Homogeneity (Scaling) | | | | | | 2. Additivity (Superposition) | +-----------------------------------+-----------------------------------+ | | Homogeneity (Scaling) | | | | | | - If you scale the input to a | | | system, the output scales by | | | the same factor | | | | | | - E.g. if doubling the input | | | voltage to a circuit result | | | in doubling the output | | | current, the circuit is | | | linear | +-----------------------------------+-----------------------------------+ | | Additivity (superposition) | | | | | | - If you input two signals into | | | the system, the output is the | | | sum of the outputs for each | | | signal individually | | | | | | - E.g. in a linear amplifier, | | | if you input two audio | | | signals, the output will be | | | the combined amplified | | | version of both signals. | +-----------------------------------+-----------------------------------+ | | **Static linearity** is where the | | | output is proportional to the | | | input by a constant factor | +-----------------------------------+-----------------------------------+ | | [Special properties of a linear | | | system: ] | | | | | | - Commutativity linear system | | | can be combined in any order | | | | | | - Superposition complex signals | | | can be broken into simpler | | | signals for analysis and | | | synthies | +-----------------------------------+-----------------------------------+ | Linear-time-invariant system | A system is linear and time | | (LTI) | variant if it satisfices two | | | additional properties | | | | | | 1. Linearity: Homogeneity and | | | Additivity | | | | | | 2. Shift invariance (Time | | | invariance) | | | | | | a. The systems behaviour | | | does not change over time | | | | | | b. E.g. A circuit behaves | | | the same today as it does | | | tomorrow. If you apply | | | the same input signal but | | | delayed by 2 seconds, the | | | output will also be | | | delayed by 2 seconds. | +-----------------------------------+-----------------------------------+ | | Why is it important? | | | | | | - Time-invariance system is | | | predictable, making them | | | easier to analyse and model. | | | | | | - If a system is a LIT then it | | | can be analysed with Fourier | | | transformation | +-----------------------------------+-----------------------------------+ | | **Examples** | | | | | | Linear and time-invariant systems | | | | | | - A resistors, capacitor, | | | inductor in a circuirt | | | | | | - An ideal amplifier | | | | | | - Wave propagation (Sound, | | | electromagnetic waves) | | | | | | - Mathematic operations like | | | differentiation, convolution | | | or recursion | | | | | | Nonlinear systems | | | | | | - A transistor operation in | | | saturation mode | | | | | | - A system where doubles the | | | input does not double the | | | output | | | | | | - Systems with thresholds, | | | saturation, hysteresis (e.g. | | | amplifiers, digital gates) | | | | | | - Voltage-power relationships | | | (P = V\^2 R) or light | | | intensity (I = e\^-aT) | | | | | | Time-variant system | | | | | | - A system whose | | | characteristics change over | | | time | | | | | | - Like a filter whose cut-off | | | frequency varies over time | +-----------------------------------+-----------------------------------+ | Synthesis | [Process of building or | | | reconstructing a signal from its | | | individual | | | components] | | | | | | In practice, **synthesis** helps | | | recreate or generate signals for | | | analysis, simulation, or testing. | | | | | | **Key concept in signal | | | analysis**: Any periodic or | | | non-periodic signal can be | | | synthesized using a sum of basic | | | waveforms | | | | | | **Key concept in system | | | analysis**: The response of a | | | system can be synthesized by | | | summing the response for an input | | | signal. This is possible for | | | *linear system.* | +-----------------------------------+-----------------------------------+ | Decomposition | Reverse of synthesis, breaking | | | down a complex signal or system | | | response into simpler components | | | for analysis | | | | | | Allowing to isolate different | | | parts of the signal, such as | | | noise or harmonics. | | | | | | **Key concept in signal | | | analysis**: Analysing a signal to | | | determine its individual | | | components. Using Fourier | | | transform to break a time-domain | | | signal into its frequency | | | components. | | | | | | **Key concept in system | | | analysis**: Break down overall | | | system behaviour into individual | | | response such as impulse response | | | (how the system reacts to a | | | single impulse) and frequency | | | response (How the system reacts | | | to inputs of varying frequencies) | +-----------------------------------+-----------------------------------+ | Superposition | Principle that the total response | | | of a system to multiple inputs is | | | the sum of the individual | | | responses to each input, when | | | assuming the system is linear | | | | | | **Key concept to signal | | | analysis**: Allows to combine | | | theses different components of a | | | signal mathematical to | | | reconstruct the signal. Analyse | | | each component separately, and | | | then sum their effects. | | | | | | **Key concept to system | | | analysis**: Property of linear | | | system, makes it possible to | | | analyse complex system behaviour | | | by breaking it into smaller | | | problems. | +-----------------------------------+-----------------------------------+ | Application of concepts | **Signal Analysis**: | | | | | | - **Synthesis**: Reconstructing | | | signals for simulations, | | | testing, or design. | | | | | | - **Decomposition**: Isolating | | | noise, harmonics, or features | | | of interest in a signal. | | | | | | - **Superposition**: | | | Understanding how different | | | signal components interact | | | (e.g., interference in | | | communication systems). | | | | | | **System Analysis**: | | | | | | - **Synthesis**: Reconstructing | | | the overall system response | | | by summing responses to | | | smaller inputs. | | | | | | - **Decomposition**: Breaking a | | | system's behavior into | | | simpler components (e.g., | | | impulse response, frequency | | | response). | | | | | | - **Superposition**: | | | Simplifying the analysis of | | | linear systems by considering | | | the response to individual | | | inputs separately. | +-----------------------------------+-----------------------------------+ | Impulse decomposition | Any signal is presented as a sum | | | (or integral) of scaled and | | | shifted impulse function. | | | | | | - Impulse functions are | | | idealized signals with | | | infinite amplitude over | | | infinitesimally short | | | duration, but with an area of | | | 1 | | | | | | - Focuses on time-domain | | | behaviour. | | | | | | - The output is determined via | | | convolution with the impulse | | | response. | | | | | | **Break signal into individual | | | samples for analysis** | +-----------------------------------+-----------------------------------+ | Step decomposition | A signal is represented as a | | | combination of scaled and shifted | | | step functions | | | | | | - Step response of a system is | | | critical in system design an | | | analysis | | | | | | - Step decomposition is used to | | | predict how a system reacts | | | to changes that occur in step | | | (e.g. switching on/off a | | | circuit) | | | | | | - Captures how a system reacts | | | to step changes, useful in | | | control system | | | | | | **Break signal into steps or step | | | functions** | +-----------------------------------+-----------------------------------+ | Fourier decomposition | **Concept**: | | | | | | - Any signal can be decomposed | | | into a sum (or integral) of | | | sinusoidal components (sines | | | and cosine) at different | | | frequencies, amplitudes and | | | phases. | | | | | | **Key Insight**: | | | | | | - Sines and cosines (or complex | | | exponentials) are the | | | **building blocks** of | | | signals. | | | | | | - Each component corresponds to | | | a specific frequency in the | | | signal. | | | | | | It is used to analyze, filter, | | | and manipulate signals in the | | | frequency domain. | | | | | | Provides a frequency-domain view | | | of signals and systems, essential | | | for understanding spectral | | | properties. | +-----------------------------------+-----------------------------------+ | Lecture 3 -- Fourier transform | | | ------------------------------ | | +-----------------------------------+-----------------------------------+ | Fourier theorem | Representing complex signals as | | | combinations of simpler sinudiod | | | signals (sine and cosine waves). | | | | | | It gives a way to analyse signals | | | in two different domains | | | | | | 1. Time domain. How the signal | | | changes over time | | | | | | 2. Frequency domain: What | | | frequency make up the signal | | | and how strong are they | | | | | | Key concepts: | | | | | | 1. **Signals as building | | | blocks** | | | | | | a. Sine and cosine waves are | | | the Lego blocks of the | | | signal | | | | | | b. A complex signals an be | | | broken down into a sum of | | | simple sine and cosine | | | waves with different | | | frequencies, amplitudes | | | and phases | | | | | | 2. **Frequency content** | | | | | | c. Every signal has energy | | | distributed across | | | different frequencies. | | | | | | d. The Fourier theorem helps | | | figure out: | | | | | | i. Which frequencies are | | | present, and how much | | | energy/strength is | | | associated with each | | | frequency | | | | | | 3. **Time and frequency | | | domains** | | | | | | e. The time domain shows how | | | a signal evolves over | | | time | | | | | | f. The frequency domain | | | shoes which frequency are | | | present and in what | | | proportions. | | | | | | g. The Fourie transforms | | | converts a signal from | | | one domain to the other | | | | | | 4. **Decomposition and | | | reconstruction** | | | | | | h. Decompose a signal to | | | find its frequency | | | components | | | | | | i. Reconstruct a signal by | | | adding all the components | | | back together = inverse | | | Fourier transform | +-----------------------------------+-----------------------------------+ | Discrete fourier transform (DFT) | DFT is a way to analyse signals | | | in the frequency domain when the | | | signal is represented by a finite | | | number of samples (discrete time | | | signals) | | | | | | - It takes a sequence of | | | sampled points from the | | | signal in the **time domain** | | | and decomposes it into a set | | | of **sinusoids** with | | | specific frequencies, | | | amplitudes, and phases. | | | | | | - The DFT provides information | | | about **which frequencies are | | | present** in the signal and | | | their corresponding | | | **magnitudes** and | | | **phases**. | | | | | | **Key Points:** | | | | | | - **Input:** A discrete-time | | | signal (finite number of | | | samples). | | | | | | - **Output:** A set of complex | | | numbers that describe the | | | amplitude and phase of the | | | frequency components. They | | | can be expressed in either | | | rectangular or polar notation | | | | | | - It's particularly useful for | | | signals sampled by digital | | | systems, like sound | | | recordings, sensor data, etc. | +-----------------------------------+-----------------------------------+ | Notation | Polar notation | | | | | | - Magnitude and phase | | | | | | - The magnitude represents the | | | amplitude of a frequency | | | components | | | | | | - The phase angle tells us the | | | offset of the sinusoid in | | | time. | +-----------------------------------+-----------------------------------+ | | Rectangular notation | | | | | | - Cartesian coordinates | | | | | | - Complex numbers with a real | | | and an imaginary part | | | | | | - The Real part is the | | | amplitudes of the cosine | | | waves | | | | | | - The imaginary parts are the | | | amplitudes of the sine waves | +-----------------------------------+-----------------------------------+ | What do we see in a DFT plot | [Magnitude spectrum] | | | | | | - **What do you see:** A plot | | | of the magnitude versus | | | frequency | | | | | | - **What does it show**: How | | | much energy the signal has at | | | each frequency. Peaks in the | | | spectrum corresponds to the | | | dominant frequencies in the | | | signal | | | | | | [Phase spectrum] | | | | | | - **What do you see:** A plot | | | of the phase angle versus | | | frequency | | | | | | - **What does it show:** The | | | relative phase (timing) of | | | the different frequency | | | components. Phase is not | | | always easy to interpret, but | | | its important to reconstruing | | | the signal correctly | | | | | | **Frequency bins** | | | | | | - The DFT divides the frequency | | | ranges into bins. Each bin | | | corresponds to a specific | | | frequency. | | | | | | - The spacing between the bins | | | depends on the sampling | | | frequency and the number of | | | points (N) in your signal. | | | | | | - Frequency resolution = fs / N | +-----------------------------------+-----------------------------------+ | Lecture 4 -- Frequency plot | | | --------------------------- | | +-----------------------------------+-----------------------------------+ | What to expect in the frequency | How does white noise look like? | | plot | | +-----------------------------------+-----------------------------------+ | | How does power line noise look | | | like? | +-----------------------------------+-----------------------------------+ | | Anti-alias filter roll-off | +-----------------------------------+-----------------------------------+ | | Signal content | +-----------------------------------+-----------------------------------+ | Fundamental frequency | The lowest frequency in a | | | periodic signal. It's the | | | frequency that determines the | | | overall repeating pattern of the | | | signal | | | | | | Key points: | | | | | | - **Periodicity**: If a signal | | | is periodic (it repeats | | | itself over time), it has | | | fundamental frequency, | | | denotes as f\_0. It's the | | | reciprocal of the period (T) | | | | | | \ | | | [\$\$f\_{0} = \\frac{1}{T}\\ | | | \$\$]{.math.display}\ | | | | | | - **Time domain**: In the time | | | domain the fundamental | | | frequency represents the | | | basic cycle of the signal | | | | | | - **Frequency domain**: In the | | | frequency spectrum, the | | | fundament frequency is the | | | typical the first peat | | | (excluding the DC) | +-----------------------------------+-----------------------------------+ | Higher harmonics | Harmonics are integer multiples | | | of the fundament frequency. These | | | represent additional frequencies | | | that contribute to the overall | | | shape of the signal | | | | | | Key points: | | | | | | - **Waveform shape**: The | | | combination of the | | | fundamental frequency and its | | | harmonics creates the signal | | | over shape | | | | | | - A pure sine wave only | | | contains fundamental | | | frequencies | | | | | | - A Square wave contains | | | both higher harmonics and | | | its fundamental | | | frequencies | +-----------------------------------+-----------------------------------+ | What defines the frequency | The frequency resolution is the | | resolution? | ability of the Fourier transform | | | to distinguish between closely | | | spaces frequency components in a | | | signal. | | | | | | Key factors: | | | | | | - **Signal duration** | | | | | | - The frequency resolution | | | improves as the duration | | | of the signal increases. | | | | | | - Resolution is defined as | | | [\$\\mathrm{\\Delta}f = | | | \\frac{1}{T}\$]{.math | | |.inline}. Where T is the | | | total time window of the | | | signal. | | | | | | - A longer signal provides | | | more detailed frequency | | | information, allowing | | | better separation of | | | close frequencies. | | | | | | - **Sampling frequency** | | | | | | - The sampling frequency | | | determines the highest | | | frequency that can be | | | analysed (the Nyquist | | | frequencies. | | | [\$f\_{\\text{nyuist}} = | | | \\frac{f\_{s}}{2}\$]{.mat | | | h | | |.inline}) | | | | | | - While fs does not | | | directly affect the | | | resolution, it dictates | | | the range of frequencies | | | visible in the DFT plot | | | | | | - **Number of points in | | | DFT (N)** | | | | | | - N affect the spacing | | | between frequency bins. | | | The bin spacing is: | | | | | | - [\$\\mathrm{\\Delta}f = | | | \\frac{f\_{s}}{N}\$]{.mat | | | h | | |.inline} | | | | | | - Where fs is the sampling | | | frequency and N is the | | | number of points in the | | | DFT. | | | | | | - Increasing N does not | | | improve the actual | | | resolution of the data, | | | but interpolates between | | | existing bins, making the | | | spectrum smoother. | +-----------------------------------+-----------------------------------+ | What is STFT | A tool to analyse non-stationary | | | signals, where the frequency | | | content changes over time. | | | | | | It dived the signal into smaller | | | overlapping segments and computes | | | the Fourier transform for each | | | segment. | | | | | | Key concepts | | | | | | Why STFT? | | | | | | - A standard Fourier transform | | | provides information about | | | which frequencies exist in a | | | signal but mot when they | | | occur. The STFT adds a time | | | dimension by analysing | | | smaller time windows. | | | | | | How does it work? | | | | | | - A windows function (e.g. | | | Hamming) is applied to | | | isolate a small segment of | | | the signal. | | | | | | - A Fourier transform is then | | | computed to the segment. | | | | | | - The process is repeated as | | | the windows slides across the | | | signal resulting in a | | | time-frequency representation | | | | | | Trade-off | | | | | | - There is a fundamental | | | trade-off between time | | | resolution and frequency | | | resolution | | | | | | - A narrow window gives better | | | time resolution but poor | | | frequency resolution | | | | | | - A wide window gives better | | | frequency resolution but poor | | | time resolution | | | | | | Output | | | | | | - The result is a spectrogram, | | | where time is one axis and | | | frequency is on other. | | | Amplitude is represented by | | | color or intensity. | +-----------------------------------+-----------------------------------+ | What Do We See in a DFT Plot? | When you look at a **DFT plot**, | | | you typically see: | | | | | | - **X-Axis (Frequency):** The | | | frequencies of the components | | | in the signal, ranging from 0 | | | to fs/2 (positive | | | frequencies) and sometimes | | | from −fs/2 to fs/2 if | | | symmetric frequencies are | | | plotted. | | | | | | - **Y-Axis (Amplitude or | | | Power):** The strength | | | (amplitude or power) of each | | | frequency component. | | | | | | - **Peaks:** Peaks in the plot | | | indicate dominant frequencies | | | in the signal. | +-----------------------------------+-----------------------------------+ | DFT of | Single cosine | | | | | | A cosine wave is: | | | [*x*(*t*) = *A*cos (2 *πf*~0~ *t* | | | )]{.math | | |.inline}, where [*f*~0~]{.math | | |.inline} is the frequency of the | | | cosine wave. | | | | | | In the frequency domain: | | | | | | - The DFT of a cosine wave | | | produces two peaks | | | | | | - One at + f\_0 (positive | | | frequency) | | | | | | - One -f\_0 (negative | | | frequency, represented | | | symmetrically) | | | | | | - The amplitude of each peak | | | corresponds to half the | | | amplitude of the cosine wave. | | | | | | Example: | | | | | | For | | | [*x*(*t*) = cos (2*π* 10 *t*)]{.m | | | ath | | |.inline} | | | | | | In the DFT, you see peaks at | | | [*f* =  ± 10 *Hz*]{.math.inline} | +-----------------------------------+-----------------------------------+ | | An impulse | | | | | | **An impulse signal** is | | | represented as: | | | | | | \ | | | [\$\$x\\left\\lbrack n | | | \\right\\rbrack = \\ \\left\\{ | | | \\begin{matrix} 1,\\ \\ n = 0 | | | \\\\ 0,\\ \\ otherwisse \\\\ | | | \\end{matrix} \\right.\\ | | | \$\$]{.math.display}\ | | | | | | In the frequency domain | | | | | | - The DFT of an impulse is | | | flat, meaning all frequency | | | components are equally | | | present. | | | | | | - The amplitude is constant | | | across all frequencies | | | | | | Why | | | | | | - An impulse contains all | | | possible frequencies with | | | equal contribution, which why | | | its spectrum is uniform | +-----------------------------------+-----------------------------------+ | | **A flat line** | | | | | | A flat/constant line is | | | represented as | | | [*x*\[*n*\] = *C*, ]{.math | | |.inline}where C is a constant | | | | | | In the Frequency domain | | | | | | - The DFT of a constant signal | | | produces a single spike at f | | | = 0 Hz (for the DC component) | | | | | | - There are no other frequency | | | components, as the signal | | | does not oscillate | | | | | | Why | | | | | | - A constant signal has no | | | variation over time, so it | | | only contains a DC component | | | (zero frequency) | +-----------------------------------+-----------------------------------+ | | | +-----------------------------------+-----------------------------------+ | Duality | ++, - - , X \* | +-----------------------------------+-----------------------------------+ | Lecture 5 -- Convolution | | | ------------------------