digital filter design video lectures

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1: The Breadth and Depth of DSP
- The Roots of DSP
- Telecommunications
- Audio Processing
- Echo Location
- Image Processing
2: Statistics, Probability and Noise
- Signal and Graph Terminology
- Mean and Standard Deviation
- Signal vs. Underlying Process
- The Histogram, Pmf and Pdf
- The Normal Distribution
- Digital Noise Generation
- Precision and Accuracy
3: ADC and DAC
- Quantization
- The Sampling Theorem
- Digital-to-Analog Conversion
- Analog Filters for Data Conversion
- Selecting The Antialias Filter
- Multirate Data Conversion
- Single Bit Data Conversion
4: DSP Software
- Computer Numbers
- Fixed Point (Integers)
- Floating Point (Real Numbers)
- Number Precision
- Execution Speed: Program Language
- Execution Speed: Hardware
- Execution Speed: Programming Tips
5: Linear Systems
- Signals and Systems
- Requirements for Linearity
- Static Linearity and Sinusoidal Fidelity
- Examples of Linear and Nonlinear Systems
- Special Properties of Linearity
- Superposition: the Foundation of DSP
- Common Decompositions
- Alternatives to Linearity
6: Convolution
- The Delta Function and Impulse Response
- Convolution
- The Input Side Algorithm
- The Output Side Algorithm
- The Sum of Weighted Inputs
7: Properties of Convolution
- Common Impulse Responses
- Mathematical Properties
- Correlation
- Speed
8: The Discrete Fourier Transform
- The Family of Fourier Transform
- Notation and Format of the Real DFT
- The Frequency Domain's Independent Variable
- DFT Basis Functions
- Synthesis, Calculating the Inverse DFT
- Analysis, Calculating the DFT
- Duality
- Polar Notation
- Polar Nuisances
9: Applications of the DFT
- Spectral Analysis of Signals
- Frequency Response of Systems
- Convolution via the Frequency Domain
10: Fourier Transform Properties
- Linearity of the Fourier Transform
- Characteristics of the Phase
- Periodic Nature of the DFT
- Compression and Expansion, Multirate methods
- Multiplying Signals (Amplitude Modulation)
- The Discrete Time Fourier Transform
- Parseval's Relation
11: Fourier Transform Pairs
- Delta Function Pairs
- The Sinc Function
- Other Transform Pairs
- Gibbs Effect
- Harmonics
- Chirp Signals
12: The Fast Fourier Transform
- Real DFT Using the Complex DFT
- How the FFT works
- FFT Programs
- Speed and Precision Comparisons
- Further Speed Increases
13: Continuous Signal Processing
- The Delta Function
- Convolution
- The Fourier Transform
- The Fourier Series
14: Introduction to Digital Filters
- Filter Basics
- How Information is Represented in Signals
- Time Domain Parameters
- Frequency Domain Parameters
- High-Pass, Band-Pass and Band-Reject Filters
- Filter Classification
15: Moving Average Filters
- Implementation by Convolution
- Noise Reduction vs. Step Response
- Frequency Response
- Relatives of the Moving Average Filter
- Recursive Implementation
16: Windowed-Sinc Filters
- Strategy of the Windowed-Sinc
- Designing the Filter
- Examples of Windowed-Sinc Filters
- Pushing it to the Limit
17: Custom Filters
- Arbitrary Frequency Response
- Deconvolution
- Optimal Filters
18: FFT Convolution
- The Overlap-Add Method
- FFT Convolution
- Speed Improvements
19: Recursive Filters
- The Recursive Method
- Single Pole Recursive Filters
- Narrow-band Filters
- Phase Response
- Using Integers
20: Chebyshev Filters
- The Chebyshev and Butterworth Responses
- Designing the Filter
- Step Response Overshoot
- Stability
21: Filter Comparison
- Match #1: Analog vs. Digital Filters
- Match #2: Windowed-Sinc vs. Chebyshev
- Match #3: Moving Average vs. Single Pole
22: Audio Processing
- Human Hearing
- Timbre
- Sound Quality vs. Data Rate
- High Fidelity Audio
- Companding
- Speech Synthesis and Recognition
- Nonlinear Audio Processing
23: Image Formation & Display
- Digital Image Structure
- Cameras and Eyes
- Television Video Signals
- Other Image Acquisition and Display
- Brightness and Contrast Adjustments
- Grayscale Transforms
- Warping
24: Linear Image Processing
- Convolution
- 3x3 Edge Modification
- Convolution by Separability
- Example of a Large PSF: Illumination Flattening
- Fourier Image Analysis
- FFT Convolution
- A Closer Look at Image Convolution
25: Special Imaging Techniques
- Spatial Resolution
- Sample Spacing and Sampling Aperture
- Signal-to-Noise Ratio
- Morphological Image Processing
- Computed Tomography
26: Neural Networks (and more!)
- Target Detection
- Neural Network Architecture
- Why Does it Work?
- Training the Neural Network
- Evaluating the Results
- Recursive Filter Design
27: Data Compression
- Data Compression Strategies
- Run-Length Encoding
- Huffman Encoding
- Delta Encoding
- LZW Compression
- JPEG (Transform Compression)
- MPEG
28: Digital Signal Processors
- How DSPs are Different from Other Microprocessors
- Circular Buffering
- Architecture of the Digital Signal Processor
- Fixed versus Floating Point
- C versus Assembly
- How Fast are DSPs?
- The Digital Signal Processor Market
29: Getting Started with DSPs
- The ADSP-2106x family
- The SHARC EZ-KIT Lite
- Design Example: An FIR Audio Filter
- Analog Measurements on a DSP System
- Another Look at Fixed versus Floating Point
- Advanced Software Tools
30: Complex Numbers
- The Complex Number System
- Polar Notation
- Using Complex Numbers by Substitution
- Complex Representation of Sinusoids
- Complex Representation of Systems
- Electrical Circuit Analysis
31: The Complex Fourier Transform
- The Real DFT
- Mathematical Equivalence
- The Complex DFT
- The Family of Fourier Transforms
- Why the Complex Fourier Transform is Used
32: The Laplace Transform
- The Nature of the s-Domain
- Strategy of the Laplace Transform
- Analysis of Electric Circuits
- The Importance of Poles and Zeros
- Filter Design in the s-Domain
33: The z-Transform
- The Nature of the z-Domain
- Analysis of Recursive Systems
- Cascade and Parallel Stages
- Spectral Inversion
- Gain Changes
- Chebyshev-Butterworth Filter Design
- The Best and Worst of DSP
34: Explaining Benford's Law
- Frank Benford's Discovery
- Homomorphic Processing
- The Ones Scaling Test
- Writing Benford's Law as a Convolution
- Solving in the Frequency Domain
- Solving Mystery #1
- Solving Mystery #2
- More on Following Benford's law
- Analysis of the Log-Normal Distribution
- The Power of Signal Processing

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Examples of Digital Filters

Digital filters are incredibly powerful, but easy to use. In fact, this is one of the main reasons that DSP has become so popular. As an example, suppose we need a low-pass filter at 1 kHz. This could be carried out in analog electronics with the following circuit:

[analog filter]

For instance, this might be used for noise reduction or separating multiplexed signals. (Chapter 3 describes how to design these analog filters). As an alternative, we could digitize the signal and use a digital filter. Say we sample the signal at 10 kHz. A comparable digital filter is carried out by the following program:

            100 'LOW-PASS WINDOWED-SINC FILTER  110 'This program filters 5000 samples with a 101 point windowed-sinc   120 'filter, resulting in 4900 samples of filtered data. 130 ' 140 '                      'INITIALIZE AND DEFINE THE ARRAYS USED 150 DIM X[4999]            'X[ ] holds the input signal 160 DIM Y[4999]            'Y[ ] holds the output signal 170 DIM H[100]             'H[ ] holds the filter kernel 180 ' 190 PI = 3.14159265 200 FC = 0.1               'The cutoff frequency (0.1 of the sampling rate) 210 M% = 100               'The filter kernel length  220 ' 230 GOSUB XXXX             'Subroutine to load X[ ] with the input signal 240 ' 250 '                      'CALCULATE THE FILTER KERNEL 260 FOR I% = 0 TO 100 270    IF (I%-M%/2) = 0 THEN H[I%] = 2*PI*FC 280    IF (I%-M%/2) <> 0  THEN H[I%] = SIN(2*PI*FC * (I%-M%/2)) / (I%-M%/2) 290    H[I%] = H[I%] * (0.54 - 0.46*COS(2*PI*I%/M%) ) 300 NEXT I% 310 '                320                        'FILTER THE SIGNAL BY CONVOLUTION 330 FOR J% = 100 TO 4999       340    Y[J%] = 0                    350    FOR I% = 0 TO 100 360       Y[J%] = Y[J%] + X[J%-I%] * H[I%] 370    NEXT I% 380 NEXT J% 390 ' 400 END

As in this example, most digital filters can be implemented with only a few dozen lines of code. How do the analog and digital filters compare? Here are the frequency responses of the two filters:

[frequency responses]

Even though we designed the digital filter to approximately match the analog filter, there are still several significant differences between the two. First, the analog filter has a 6% ripple in the passband, while the digital filter is perfectly flat (within 0.02%). The analog designer might argue that the ripple can be selected in the design; however, this misses the point. The flatness achievable with analog filters is limited by the accuracy of their resistors and capacitors. Even if it is designed for zero ripple (a Butterworth filter), analog filters of this complexity will have a residue ripple of, perhaps, 1%. On the other hand, the flatness of digital filters is primarily limited by round-off error, making them hundreds of times flatter than their analog counterparts.

Next, let's look at the frequency response on a log scale (decibels), as shown below. Again, the digital filter is clearly the victor in both roll-off and stopband attenuation .

[frequency responses]

Even if the analog performance is improved by adding additional stages, it still can't compete with the digital filter. Imagine you need to improve the performance of the filter by a factor of 100. This would be virtually impossible for the analog circuit, but only requires simple modifications to the digital filter. For instance, look at the two frequency responses below, a digital filter designed for very fast roll-off, and a digital filter designed for exceptional stopband attenuation.

[high performance filters]

The frequency response on the left has a gain of 1 +/- 0.0002 from DC to 999 hertz, and a gain of less than 0.0002 for frequencies above 1001 hertz. The entire transition occurs in only about 1 hertz. The frequency response on the right is equally impressive: the stopband attenuation is -150 dB, one part in 30 million! Don't try this with an op amp!

As in these examples, digital filters can achieve thousands of times better performance than analog filters. This makes a dramatic difference in how filtering problems are approached. With analog filters, the emphasis is on handling limitations of the electronics, such as the accuracy and stability of the resistors and capacitors. In comparison, digital filters are so good that the performance of the filter is frequently ignored. The emphasis shifts to the limitations of the signals, and the theoretical issues regarding their processing.

[custom filter]

Here is another example of the tremendous power of digital filters. Filters usually have one of four basic responses: low-pass, high-pass, band-pass or band-reject. But what if you need something really custom? As an extreme example, suppose you need a filter with the frequency response shown at the right. This isn't as far fetched as you might think; several area of DSP routinely use frequency responses this irregular (deconvolution and optimal filtering). Don't ask an analog filter designer to give you this frequency response- he can't! In comparison, digital filters excel at providing these irregular curves. And the best part: The Scientist and Engineer's guide to DSP tells you how to create these filters quickly and easily.