Digital Signal Processing -EC504- Module2 ( MAKAUT-Syllabus)

 Today, our communication elements are:  Design of FIR Digital Filters, Window method, Parks-McClellan Method, Design of IIR Digital Filters, Butterworth filter, Chebyshev and Elliptic Approximations, Low-Pass Filters, Band pass filter, Band-Stop Filters, High Pass Filter, Difference Between Low-Pass, High-Pass, Band-Pass, and Band-Stop Filters.

Design of FIR Digital Filters: -

The final impulse response (FIR) filter is an essential class of digital filters widely used in signal processing due to their stability and linear step properties. The design of the FIR filter involves determining the filter coefficient so that the filter comes from the desired frequency response specifications, such as passband, stopband tape and transition width.

A four-filter is defined by its impulse response, which consists of a limited period as opposed to the IIR filter. Production is expressed in this way:

where $h[k]$ represents the filter coefficients, and $M$ is the filter order. The design process focuses on selecting $h[k]$ to achieve the required frequency response.

There are many ways to design FIR filters:

1. Window Method – this is the simplest technique, where an ideal endless impulse response is cut through a window function such as Hamming, male, or Blackman. The window's alternative affects the performance of the filter in the context of the infection bandwidth and the ripple.

2. Frequency sampling method – In this approach, the desired frequency response is tested directly on the frequency spots on the site, and the filter coefficient is calculated using the inverse discrete Fourier transformation (IDFT).

3. Optimal design methods – highly advanced techniques, such as Park-Kaklalan algorithms, reduce the maximum failure between desired and real frequency reactions, producing an optimal filter for the given specifications.

The FIR filters are particularly valuable for their linear phase action, which ensures no phase deformation in signals such as audio or data communication. They are widely used in applications such as speech processing, image growth, biomedical signal analysis, and wireless communication.

Window method: -

The window method is one of the most commonly used techniques to design digital filters (FIR) digital filters. It is simple and practical, and provides good control over trade ties between filter performance and calculation complexity.

The basic idea of the window method is to begin with the ideal impulse response to the desired filter, which is usually infinite during the period, and then it is cut to a limited length using a window function. By multiplying the ideal response by a selected window, we get a real four filter with a limited number of coefficients.

Mathematically, if $h_d[n]$ is the ideal impulse response and $w[n]$ is the window function; the FIR filter is given by: h[n]=hd​[n]⋅w[n].

Different window features are available, with each having unique properties. The most common are:

• Rectangular window – the simplest shape, but produces vital waves (Gibbs phenomenon).
• Harming reduces levels of the window side and offers better stopband orthopedics.
• Hanging window – smooth infection, but with a slightly wide main tag.
• Blackman Window – Provides high stopping commercials at the expense of wide inflection areas.

The choice of window depends on the requirements for the width of the main lobe (inflection sharpness) and the requirements for side locking level (ripple control).

The window method is widely used due to its simplicity and ease of implementation. Although it does not always provide the most optimal filter, it provides satisfactory results for many real-world applications, such as audio processing, image filtration, and data equalization.

Parks-McClellan Method: -

The Parks-McClellan method, also known as the equipped method, is a widely used algorithm to design the optimal final impulsive reaction filter (FIR). Unlike simple approaches such as the window method, which depends on the rest of the ideal response, the park-cladal algorithm optimizes the direct filter coefficient to obtain the best possible frequency response for the specifications provided directly.

The method is based on the principle of minimax adaptation, which reduces the maximum error between the desired and real frequency responses. As a result, the error is evenly distributed in passenger tape and stop tape and provides a similar feature – hence the name.

The Parks-McClellan method provides many benefits:

• Optimal filter design for a given filter order and specifications.
• Make sure of linear phase action; make sure there is no phase deformation.
• Flexibility in the design of smaller passports, high passes, band passes, and band-stop filters.

However, the method can be more intensive than simple approaches, and it requires accurate specifications for passenger strip and stopping orthopedics.

Despite these challenges, the Parks-McClellan method is one of the most powerful techniques for FIR filter design. Much is used in audio equipment, communication systems, biomedical signal analysis, and image processing, where high filter accuracy is needed.

In summary, the Parks-McLaughlin method provides an optimal and systematic approach to FIR filter design, making it a foundation stone in modern digital signal processing.

Design of IIR Digital Filters: -

An analog impulse response (IIR) is widely used in digital filters and digital signal processing due to its efficiency and ability to cause analog filter capacity with relatively low filter order. Unlike the FIR filter, which contains a final impulse reaction, the IIR filter consists of feedback paths, leading to its impulse response to endless expansion.

The design of the IIR filter often begins by converting a well-known analog filter (such as a Butterworth, Chebyshev, or elliptical filter) to a digital domain. This process ensures that the desirable properties of the analog prototype are preserved. There are two general change techniques:

Impulse irreversible changes - maps the impulse response to an analog filter in digital domains but may be the cause of aliasing.

Bilinear transformation – The frequency blows the axis to avoid lighting and ensures stability, making it the most commonly used method.

The IIR filter can achieve sharp frequency selectivity with a lower coefficient than the four filters, making them calculate effectively. However, they may suffer from non-linear phase action, which can distort the signals where the stages are important.

Applications of the IIR filter include audio equipment, biomedical signal filtering, voice treatment, and communication systems. Their balance between efficiency and performance makes them an essential tool in real-time signal treatment.

Butterworth filter: -

The Butterworth filter is one of the most popular types of analog and digital filters, which is widely used in signal processing due to the maximum flat frequency response in the passband. Unlike other filters that present waves in the passenger tape or stopband, the Butterworth filter provides a smooth and monotonous reaction, making it ideal for applications where signal deformation needs to be minimised.

The N-Order Butterworth is given by the quantitative reaction of the low-pass filter:

Here, 𝜔𝑐 is a cutting frequency, and n is a filter order. As the filter order increases, the isolation between the passband and the stopband increases, providing better frequency selectivity. However, high-order filters require more calculation resources and can introduce stability challenges in practical designs.

Butterworth filters can be applied as low-pass, high-pass, band-pass, or tape filters on the basis of application. They are usually designed in the first analog domains and are then converted to digital form using changes such as a bilinear transformation or an impulsive Invariant method.

A great advantage of the Butterworth filter is the absence of waves, unlike Chebyshev or elliptical filters. However, at the expense of a slow rollout in the transition area, compared to these options.

Applications of Butterworth filters include sound treatment, where even frequency reaction ensures natural sound; biomedical signal analysis, such as ECG filtration; and communication systems, where noise suppression and signaling separation are important.

In summary, the Butterworth filter gives an excellent trade-off between simplicity, robustness, and performance, making it one of the most widely used filters in digital and analog signal processing.

Chebyshev and Elliptic Approximations: -

In digital filter design, Chebyshev and elliptical filters are used when two advanced estimates are used when a sharp inflection between the passband and stopband is required, compared to the Butterworth filter. Both provide high selectivity, but introduce the waves of the frequency reaction as a business band.

Chebyshev -Filters come in two types:

• Type I has similar behavior in the passenger tape and a monotonous stop tape.
• Type II (also known as the reverse Chebyshev) consists of a flat passband and an quirpele behavior in the stopband.

The Chebyshev filters receive a sharp roll-out compared to the Butterworth filter of the same order, which means that they require smaller coefficients for equal level selectivity. However, the passenger tape or stopband can introduce either ripple deformities, which should be considered in applications where signal accuracy is important.

The elliptical filter, also known as the Causer filter, is even more selective. They perform waves in both the passenger tape and the stopband, but provide the fastest infection for a given filter order. This means that they need minimal calculation efforts to achieve strict frequency specifications. Trading is that the presence of the wave in both regions may not be suitable for applications that seek very smooth reactions.

Mathematically is both obtained from an intelligent Chebyshev polynomial, and its design involves controlling ripple forms and cutting frequencies.

In practice, Chebyshev and elliptical filters are widely used in telecommunications, radar, and sound systems, where effective filtration with dense specifications is required.

In summary, the chill-lift selectivity balances with limited ripple, while elliptical filters provide the most effective transition, but at the expense of the wave in both bands.

Low-Pass Filters: -

A low-pass filter (LPF) is one of the most commonly used filters in signal processing and communication. The primary function is to allow the low existing components of a signal to be carried when you reduce or block high existing components. This creates an essential tool for removing noise, signal leveling, and useful information from raw signals.

The frequency at which the filter infection from the passband to the stopband is known as a cut-off frequency. The frequencies below this point are preserved with little or no odorization, while the frequencies above it are suppressed. The stability of this transition depends on the design and order of the filter.

Smaller pass filters can be used in both the analog and digital domains. In the analog circuit, they are often made using resistance, capacitors, or operational amplifiers. In the digital system, they are labeled using a FIFE (final impulse response) or IIR (infinite impulse response) structures, such as algorithms.

Many ornate techniques are used to design low-pass filters, including Butterworth (maximum flat response), Chebyshev (equiripple properties), and elliptical filters (most infections to waves in both bands). The alternative depends on applications such as lubrication, sharpness, or calculation efficiency.

Low-pass filters have different applications. In sound treatment, they eliminate high-frequency noise to produce cleaner sound. In image processing, they lubricate images by removing sharp edges or noise. In communication, they distinguish the baseband signal from high existing noise and interference. Even in biomedical design, they are used to filter ECG or EEG signals for accurate analysis.

Band pass filter: -

A band pass filter (BPF) is a basic signal processing tool known as a pass filter to reduce frequencies outside this area, allowing indications within a specified frequency range. Unlike low-pass or high-pass filters, which work on the edge of the spectrum, a band-pass filter defines both a low cut-off frequency (FL) and an upper cutting frequency (FH). The signals within these limits are preserved, while the indications of the FH are significantly reduced.

Types of band-pass filter

1. The analog band pass filter-dienic components (resistance, capacitors, inductors) or active components (operating amplifiers). These are widely used in radio receivers and sound systems.

2. Digital Band pass filter-PAS filter-FIR or IIR structures were implemented using algorithms. They are used on flexible, reconstructive, and often software-defined radio, DSP systems, and biomedical devices.

Design approach:

• Butterworth band-pass filter: provides a maximum flat response in the passband, which ensures smooth signal transmission without waves.
• Chebyshev Band-Pass filter: Depending on the type, sharp cuts receive waves in the passenger straps or stopbands.
• Elliptical tape pass filter: provides the narrowest transitional tape for a given filter order, but introduces waves in both bands.
• Window and FFT-based digital methods: Useful in the design of practical FIR-band-pass filters with controlled frequency responses.

Application of band-pass filter:

1. Communication system-tape-pas filters are required for setting a specific channel frequency when you reject others. In radio, for example, they distinguish the desired broadcast frequency from the entire range of signals.

2. Sound treatment - they help remove or increase some frequency range, such as separating vocal frequencies in music production.

3. Biomedical engineering- filters are used in ECG and EEG analysis to focus on clinically relevant frequency bands by removing the noise.

4. Radar and sonar - they distinguish indications of interest from unwanted echoes or intervention.

5. Instrumentation and measurement - the noise is used to detect weak signals in noise by focusing on a narrow frequency band.

conclusion

A band-pass filter is an indispensable tool in modern signal processing, which affects a balance between choosing useful signals and rejecting unwanted noise. Defining both lower and upper cutting frequencies enables accurate control over the frequency material of the signals. Whether used in an analog circuit or digital system, band-pass filters play an important role in communication, medical equipment, sound technology, and scientific equipment.

Band-Stop Filters: -

A band-stop filter (BSF), also known as a tape rejection filter or notch filter (when the rejected tape is very narrow), is an important tool in signal processing that opposes a tape pot filter. Instead of allowing a certain range of frequencies to pass, reflect a tape-stop filter, or block a specific frequency band so that the frequencies can pass the frequencies outside this area. This makes it ideal to eliminate unwanted intervention or noise in a character and preserve the useful frequency components.

Types of tape stop filter

1. Analog band-stop filter surgery was implemented using inactive elements (resistance, capacitors, inductors) or an active cycle with amplifiers. Common in radio receivers and sound devices.

2. The digital band-stop filter-DSP system designed with FIR or IIR structures. They are very flexible and can be compatible with different rejection bands as needed.

Design approach

1. Butterworth Band-Stop: Passband has no wave, but a smooth reaction with gradual roll-out.

2. Chebyhev Band-Stop: Sharper Transition, but introduces waves in either lower or higher passenger tape.

3. Elliptical tape stop: Provides a narrow transition area, but both passages include waves.

4. Window Method (Digital Fire): Customizable, remarkable filters for accurate digital applications is used to design.

Application of the band stop filter:

1. Removal of streamlined intervention - in biomedical devices such as ECG and EEG, the Notch filter is used at 50 Hz or 60 Hz to eliminate the noise from the power supply while retaining the physical signals.

2. Sound treatment - typical unwanted frequencies, such as hum or reaction, are used to suppress noise, while maintaining the overall sound quality.

3. Communication system - Communication helps reject frequencies or jam signals in a ribbon without disturbing the rest of the spectrum.

4. Instrumentation - useful in measurement systems to eliminate specific intervention signals and increase accuracy.

5. Radar and sonar systems - remove the echoes that make strong interference at specific frequencies to improve the ability to detect.

conclusion

The tape-stop filter is a powerful tool for choosing unwanted frequency components and preserving the useful parts of a signal. The ability to "exclude" noise or intervention is indispensable in areas ranging from biomedical engineering science to communication and sound treatment. Whether digitally designed with a circuit in analog form or through an algorithm, tape-stop filter signals ensure clarity and increase performance in countless applications. Their versatility and accuracy of rejecting targeted frequencies make them one of the most valuable filters in modern signal processing.

High Pass Filter: -

A high-pass filter (HPF) signal is a basic tool in processing and electronics that makes it possible to pass high existing characters while reducing low existing signals. Essentially, it blocks or reduces the components under a certain area, called the cutting frequency, and allows characters above this frequency with minimal damping. High-pass filters are widely used in communication systems, sound engineering, biomedical instruments, and digital signal processing.

Types of high-pass filter

1. Analog was implemented by using active components such as high-pass filter resistance, capacitors, and inductors (RC, RL, or RLC circuits) or operational amplifiers.

2. The digital high-pass filter-DSP system allows FIR (final impulse response) or IIR (infinite impulse response) algorithms, allowing accurate control and recycling.

Common Design Approaches:

1. Butterworth high-pass filter: Provides a smooth frequency response without a wave, which ensures maximum flat pass band.

2. Chobyhev High-Pass Filter: Provides stator drain compared to Butterworth, but introduces waves in the passenger tape.

3. Elliptic high-pass filter: provides the fastest roll-out for a given filter order, but is wavy in both the passband and stopband.

4. Window Method (FIR filter): Common in digital systems to design high-pass filters with desired frequency properties.

High-pass filter application

1. Audio Engineering –Protecting highly existing components such as sound engineering vocals and instruments, which we rumble or a microphone to remove noise handling.

2. Communication Systems –The purpose of the communication system as indications of a certain frequency by eliminating interference and deformation with low existence.

3. In biomedical engineering, EEG signal therapy eliminates high-pass filter baseline operation due to slow physical changes, which improves signal clarity.

4. An image processing high-pass filter can increase the edges and fine details by pressing out slow variations in the brightness.

5. Instrumentation-Removes operation or DC components with low existence in measurement systems to focus on characters.

conclusion

High-pass filters are indispensable in modern signal treatment, which plays an important role in separating high-frequency components that are useful from unwanted low-frequency noise or deformation. Their versatility extends to analogue and digital implementation, such as Butterworth, cheese, or elliptical responses to fit specific requirements. Whether used in sound systems, communication, medical equipment, or digital imaging, a high-pass filter enables clarity, accuracy and efficiency in handling signals in the real world.

By controlling which frequencies are allowed and which are suppressed, high-pass filtering engineers and researchers help focus the most relevant information, making them one of the corners of signal processing and electronics.

Difference Between Low-Pass, High-Pass, Band-Pass, and Band-Stop Filters: -

Feature	Low-Pass Filter (LPF)	High-Pass Filter (HPF)	Band-Pass Filter (BPF)	Band-Stop Filter (BSF) (Notch Filter)
Definition	Passes low frequencies below a cutoff frequency and attenuates high frequencies.	Passes high frequencies above a cutoff frequency and attenuates low frequencies.	Passes a specific range (band) of frequencies and attenuates frequencies outside this range.	Attenuates or rejects a specific range (band) of frequencies and allows frequencies outside this range.
Cutoff Frequency (fc)	One cutoff frequency (fc) that separates the passband and stopband.	One cutoff frequency (fc) that separates the stopband and passband.	Two cutoff frequencies: lower (fL) and upper (fH), defining the passband.	Two cutoff frequencies: lower (fL) and upper (fH), defining the stopband.
Passband	Frequencies below fc.	Frequencies above fc.	Frequencies between fL and fH.	Frequencies below fL and above fH.
Stopband	Frequencies above fc.	Frequencies below fc.	Frequencies below fL and above fH.	Frequencies between fL and fH.
Applications	Audio equalizers, anti-aliasing, and smoothing signals.	Audio noise removal, communication systems, and biomedical drift removal.	Radio receivers, audio processing, and biomedical instrumentation.	Power line interference removal, communication noise suppression, and instrumentation.
Example Use Case	Removing high-frequency noise from ECG signals.	Removing low-frequency rumble from microphones.	Isolating a particular radio channel.	Eliminating 50/60 Hz power line interference in biomedical devices.

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