Signals II (DSP Fundamentals) - 5ESC0

5ESC0 Signals II (DSP Fundamentals)

• Area: Signals and Systems
• Topics: Fourier Transform · Z-Transform · Sampling · Transfer Functions and Frequency Response · Filters · Random Signals and Power Spectral Density
• Related courses: Systems - 5ESB0 · Control Systems - 5ESD0 · Communication 1 - 5ETC0
• Notes by a friend: Fourier Analysis (friend) - 5ESC0 · Laplace Transform (friend) - 5ESC0 · Overview (friend) - 5ESC0 · Prerequisites (friend) - 5ESE0 · Transforms (friend) - 5ESC0 · Z-Transform (friend) - 5ESC0

Pre-knowledge

Introduction_5ESC0 (4).pdf


Geometric series - Wikipedia

rule of thumbs: zero refers to the function (e.g. polynomial) and root refers to the equation.
3.6: Complex Zeros - Mathematics LibreTexts

Signals and System

Signals_and_Systems (1).pdf

Convolution:
Frequency response:






linearity: think of it in root locus plot: if line is continuous no jumps: linear

delayed impulses → convolution
complex exponentials → Fourier analysis

Fourier Analysis

Topic: Fourier Transform

Fourier Analysis (friend) - 5ESC0
Fourier Analysis_Peri (1).pdf

Continuous Time Fourier Transform (CTFT)

1. Forward Fourier Transform:

2. Inverse Fourier Transform:

Fourier Series for Periodic Signals

1. Fourier Series Representation:

2. Fourier Series Coefficients:

Fourier Transform of Discrete-Time Signals (FTD)

1. Forward Discrete-Time Fourier Transform (DTFT):

2. Inverse Discrete-Time Fourier Transform (IDTFT):

Common FTD Pairs

• Delta function in time:

• Exponential sequence:

Frequency Response of an LTI System

1. Impulse Response and Frequency Response Pair:

2. Convolution in Time Domain:

Sampling

Topic: Sampling
TUe_Signals2_SamplingInterpolation2020.pdf

Convolution Integral

1. Convolution in Time Domain:

2. Convolution Properties:

   • Time domain convolution corresponds to multiplication in frequency domain .
   • Multiplication in time domain corresponds to convolution in frequency domain .

Fourier Transform of a Dirac Train (Pulse Train)

1. Dirac Train in Time Domain:

2. Corresponding Fourier Transform in Frequency Domain:

Nyquist Sampling Theorem

To avoid aliasing, the sampling rate should be:

Discrete Fourier Transform (DFT)

1. DFT Expression:

2. Inverse DFT:

Ideal Low Pass Filter

1. Time Domain:

2. Frequency Domain:

Interpolation Formula

For reconstructing a continuous signal from samples :

where

The DFT

Topic: Fourier Transform
The DFT_2022_Peri.pdf

Discrete Fourier Transform (DFT)

1. DFT Definition:

2. Inverse DFT (IDFT):

Twiddle Factor

1. Twiddle Factor Definition:

2. Property of Twiddle Factor:

DFT Properties

1. Linearity:

2. Symmetry for Real Sequences:

3. Circular Shift:

   • Time Shift:
   • Frequency Shift:

Circular Convolution

1. Circular Convolution in Time Domain:

2. Circular Convolution Property:

   • Convolution in the time domain corresponds to multiplication in the frequency domain:

Zero Padding

1. Zero Padding:

   • Extending a signal with zeros to increase DFT resolution without changing signal content.
   • With zero padding, the DFT length increases, and frequency resolution improves.

2. Zero Padding and DFT Resolution:

   • For , frequency bins are apart.
   • For , frequency bins are apart.

Practical Applications

1. Sampling in DFT Domain:
   • Sampling a continuous-time Fourier Transform (CTFT) creates repetitions in the DFT, commonly leading to spectral leakage if not handled with appropriate windowing.
2. Windowing:
   • Window functions like Hanning window:
   • Windowing minimizes edge effects and reduces spectral leakage.

Z-transform

Topic: Z-Transform
Z-transform.pdf

Z-Transform Definition

1. Z-Transform:

2. Inverse Z-Transform:

Z-Transform Properties

1. Linearity:

2. Time Shift:

3. Scaling in the z-Domain:

4. Convolution:

   • Convolution in time domain is multiplication in the z-domain:

Region of Convergence (ROC)

• For a right-sided sequence (causal):
• For a left-sided sequence:
• For a two-sided sequence, ROC is a ring:

Common Z-Transform Pairs

1. Delta Function:

2. Shifted Delta Function:

3. Exponential Sequence:

4. Cosine Sequence:

Examples of Z-Transforms

1. For :

2. For :

Z-Transform of Convolution

1. Convolution in the Time Domain:

   where ( * ) denotes convolution.

2. Example with Exponential Input:

   • For and :
   • Then, .

Inverse Z-Transform (Partial Fraction Expansion)

1. Example:
   Given: Decompose as: Solve for constants ( A ) and ( B ), and apply inverse Z-transform.

Summary of Important Concepts

• The Z-transform provides a method for analyzing discrete-time signals and systems, particularly useful for characterizing stability and frequency response.
• The ROC determines the stability and applicability of the Z-transform for different signals.
• Common Z-transform pairs and properties allow quick transformations between time and z-domains.

System function

Topic: Transfer Functions and Frequency Response
System_function.pdf

System Function Definition

1. System Function (Z-transform of impulse response):

2. Frequency Response:

LTI System Properties

1. Convolution Property:

   • Output as a convolution of input with impulse response :
   • In the Z-domain, this corresponds to multiplication:

2. System Function Expression:

Example System Function

1. For a system with difference equation:
   • Rearrange to find :

Poles and Zeros

• Zeros are the roots of the numerator polynomial.
• Poles are the roots of the denominator polynomial.

Stability and Causality

1. Stability (Bounded Input, Bounded Output - BIBO):

   • System is stable if the sum of impulse response values is finite.
   • ROC (Region of Convergence) must include the unit circle for stability.

2. Causality:

   • A causal system has a right-sided impulse response.
   • For a causal system, ROC is outside the outermost pole:

Example: FIR and IIR Filters

1. FIR (Finite Impulse Response) Filter:

   • Poles are located at .

2. IIR (Infinite Impulse Response) Filter:

   • Poles are within the ROC, but not limited to .

All-Pass System

1. Pole-Zero Relationship for All-Pass Filters:
   • Zeros are reflected across the unit circle to obtain poles.
   • If , then the mirrored pole is:

Minimum and Maximum Phase

• Minimum-phase system: All zeros are inside or on the unit circle.
• Maximum-phase system: Zeros are outside the unit circle.

Filter structure

Topic: Filters
8 - Filter_structure.pdf

FIR Filter Structure

1. FIR Filter:
   • Finite Impulse Response (FIR) filter has no feedback loops, meaning it has no poles except at and .
   • The FIR filter output can be expressed as:
   • Can be implemented using a delay line and multiply-add structure:
     • Each delayed signal ) is multiplied by and summed to produce .

Linear Phase Filters

1. Linear Phase Condition:

   • A filter has linear phase if its impulse response satisfies: where ( N ) is the filter order.
   • The phase shift is linearly dependent on frequency , ensuring that all frequencies are delayed by the same amount.

2. Impulse Response Symmetry:

   • For odd ( N ): symmetry axis coincides with a sample .
   • For even ( N ): symmetry axis lies between two samples.

Types of Linear Phase FIR Filters

1. Type I: Symmetric impulse response, odd length.

   • No restrictions on filter type (can be used as low-pass, high-pass, band-pass, or band-stop).

2. Type II: Symmetric impulse response, even length.

   • Has a zero at ( z = -1 ) (cannot be used as a high-pass filter).

3. Type III: Antisymmetric impulse response, odd length.

   • Has zeros at , suitable for band-pass but not low-pass or high-pass.

4. Type IV: Antisymmetric impulse response, even length.

   • Has a zero at ( z = 1 ), not suitable for low-pass filters.

Frequency Response of Linear Phase FIR Filters

1. For Type I FIR Filter:

   • Can be used to design any filter type without restrictions.

2. For Type II FIR Filter:

   • Cannot be used to design a high-pass filter.

3. For Type III FIR Filter:

   • Cannot be used for low-pass, high-pass, or band-stop due to zeros at .

4. For Type IV FIR Filter:

   • Not suitable for low-pass filters as it has a zero at ( z = 1 ).

Key Properties of FIR Filters

1. FIR filters with linear phase have the advantage of maintaining the shape of signals within the passband.
2. They are commonly implemented as non-recursive structures with a finite number of taps (coefficients ( h[n] )).
3. Filters are stable as they do not rely on feedback, meaning there is no risk of poles moving outside the unit circle.

Summary

• FIR filters can be implemented in various forms (transversal, cascade, etc.) based on application requirements.
• Linear phase is only possible with FIR filters and requires specific symmetry in the impulse response.
• Different types of linear phase FIR filters have distinct applications based on their frequency response characteristics.

Filter design

Topic: Filters
9 - Filter Design.pdf

Ideal Filter Examples

1. Ideal Low-Pass Filter:

   where is the cutoff frequency.

2. Ideal High-Pass Filter:

3. Ideal Band-Pass Filter:

4. Ideal Band-Stop Filter:

Filter Design Specifications

1. Passband Ripple : Allows a specified maximum ripple in the passband.
2. Stopband Attenuation : Defines the minimum attenuation level in the stopband, often given in dB.

FIR Filter Design Process (Windowing Method)

1. Filter Length:

   • The filter length ( N ) is chosen to control the filter’s main lobe width and transition width.
   • As ( N ) increases, the main lobe narrows, decreasing the transition width.

2. Window Types and Characteristics:

   • Rectangular Window:
     • Main lobe width
     • Peak sidelobe level: -13 dB.
   • Hanning Window:
     • Main lobe width
     • Peak sidelobe level: -31 dB.
   • Hamming Window:
     • Main lobe width
     • Peak sidelobe level: -41 dB.
   • Blackman Window:
     • Main lobe width
     • Peak sidelobe level: -57 dB.

3. Filter Coefficients:

   • Impulse response ( h[n] ) for a low-pass filter:

FIR Filter Design Using Frequency Sampling

1. Desired frequency response is sampled at uniformly spaced frequencies.
2. FIR filter coefficients ( h[n] ) are obtained via Inverse Discrete Fourier Transform (IDFT).

Equiripple Linear Phase FIR Filter

1. Allows ripple to be uniformly distributed across the passband and stopband, producing a smaller peak ripple than non-equiripple methods.

Stochastic signals

Topic: Random Signals and Power Spectral Density
Stochastic Signals.pdf

Stochastic Process Basics

1. Ensemble Average (Expected Value):

   where is the probability of occurrence of the random variable .

2. Mean Estimate:

Second-Order Statistics

1. Mean:

2. Variance:

3. Autocorrelation:

4. Autocovariance:

5. Cross-Correlation (between two processes and ):

6. Cross-Covariance:

Wide Sense Stationarity (WSS)

• A process is WSS if:
  • Mean is constant over time.
  • Autocorrelation depends only on the lag , not on .

Ergodicity

1. Time-Averaged Mean for Ergodic Signals:

2. Time-Averaged Variance:

3. Autocorrelation for Ergodic Signals:

Power Spectral Density (PSD)

• Wiener-Khinchin Relation:
• The average power:

Example PSD Calculation

1. For with :