2.5 Finite impulse response (FIR) filters

FIR filters are digital filters with a finite impulse response, known for their linear phase and stability. They're crucial in signal processing, offering precise control over frequency responses. FIR filters find applications in audio, image, and biomedical signal processing.

These filters are designed using methods like windowing and Parks-McClellan algorithm. They can be implemented through direct form or transposed structures. While FIR filters offer advantages like stability and linear phase, they may require higher computational resources compared to IIR filters.

Properties of FIR filters

• Finite Impulse Response (FIR) filters are digital filters that possess a finite impulse response, meaning their output settles to zero in a finite number of sample intervals when an impulse is applied to the input
• FIR filters are characterized by their linear phase response, stability, and causality, which make them suitable for various signal processing applications in the context of Advanced Signal Processing
• The properties of FIR filters are determined by the filter coefficients, which are the values that define the impulse response of the filter

Linear phase response

• FIR filters can be designed to have a linear phase response, which means that the phase shift introduced by the filter is a linear function of frequency
• Linear phase response is crucial in many applications, such as audio and speech processing, where preserving the waveform shape and avoiding phase distortion is important
• To achieve a linear phase response, the impulse response of the FIR filter must be symmetric or antisymmetric about its midpoint
• Symmetric impulse response: $h[n] = h[N-1-n]$, where $N$ is the filter length
• Antisymmetric impulse response: $h[n] = -h[N-1-n]$

Stability of FIR filters

• FIR filters are inherently stable because they have no feedback and their impulse response is finite in duration
• The stability of FIR filters is guaranteed regardless of the filter coefficients, as long as the coefficients are real and finite
• The absence of feedback in FIR filters eliminates the possibility of unstable oscillations or divergence, which can occur in Infinite Impulse Response (IIR) filters with poorly chosen coefficients

Causality in FIR filters

• FIR filters are causal systems, meaning that their output depends only on the current and past input samples, not on future samples
• Causality is an important property in real-time signal processing applications, where the filter output must be computed based on the available input samples
• The impulse response of a causal FIR filter satisfies the condition: $h[n] = 0$ for $n < 0$, where $n$ represents the time index

Impulse response of FIR filters

• The impulse response of an FIR filter is the output of the filter when an impulse signal is applied to its input
• The impulse response fully characterizes the behavior of the FIR filter and determines its properties, such as frequency response and phase response

Finite duration

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• The impulse response of an FIR filter has a finite duration, meaning it settles to zero after a certain number of sample intervals
• The length of the impulse response, denoted as $N$, determines the order of the FIR filter, which is $N-1$
• A longer impulse response allows for better frequency selectivity and stopband attenuation but increases the computational complexity and delay of the filter

Transfer function of FIR filters

• The transfer function of an FIR filter is the z-transform of its impulse response, given by: $H(z) = \sum_{n=0}^{N-1} h[n] z^{-n}$
• The transfer function represents the relationship between the input and output of the filter in the z-domain
• The coefficients of the transfer function are the same as the impulse response values, $h[n]$

Zeros of transfer function

• The zeros of the transfer function are the values of $z$ for which $H(z) = 0$
• FIR filters have only zeros in their transfer function, as opposed to IIR filters, which have both zeros and poles
• The location of the zeros in the z-plane determines the frequency response of the FIR filter
• Zeros located close to the unit circle in the z-plane result in sharp transitions in the frequency response, while zeros far from the unit circle have a less pronounced effect

Difference equation for FIR filters

• The difference equation describes the input-output relationship of an FIR filter in the time domain
• For an FIR filter of order $N-1$, the difference equation is given by: $y[n] = \sum_{k=0}^{N-1} h[k] x[n-k]$
• The output sample $y[n]$ is computed as a weighted sum of the current and past input samples, $x[n], x[n-1], ..., x[n-N+1]$, where the weights are the filter coefficients, $h[k]$

Convolution and FIR filters

• The output of an FIR filter can be computed by convolving the input signal with the filter's impulse response
• Convolution is a mathematical operation that combines two signals to produce a third signal, representing the output of a linear time-invariant (LTI) system
• The convolution of the input signal $x[n]$ with the impulse response $h[n]$ is given by: $y[n] = x[n] * h[n] = \sum_{k=0}^{N-1} h[k] x[n-k]$
• Convolution in the time domain is equivalent to multiplication in the frequency domain, which allows for efficient implementation of FIR filters using the Fast Fourier Transform (FFT) algorithm

Frequency response of FIR filters

• The frequency response of an FIR filter describes how the filter modifies the amplitude and phase of the input signal as a function of frequency
• The frequency response is obtained by evaluating the transfer function $H(z)$ on the unit circle, i.e., $z = e^{j\omega}$, where $\omega$ is the angular frequency in radians per sample

Magnitude response

• The magnitude response represents the gain of the filter as a function of frequency
• It is computed as the absolute value of the frequency response: $|H(e^{j\omega})| = \sqrt{\text{Re}(H(e^{j\omega}))^2 + \text{Im}(H(e^{j\omega}))^2}$
• The magnitude response determines the passband and stopband characteristics of the filter, such as the cutoff frequency, transition bandwidth, and stopband attenuation

Phase response

• The phase response represents the phase shift introduced by the filter as a function of frequency
• It is computed as the argument of the frequency response: $\angle H(e^{j\omega}) = \arctan\left(\frac{\text{Im}(H(e^{j\omega}))}{\text{Re}(H(e^{j\omega}))}\right)$
• For FIR filters with linear phase response, the phase response is a linear function of frequency, with a constant group delay

FIR filter design methods

• Various methods exist for designing FIR filters with desired frequency response characteristics
• These methods aim to determine the filter coefficients that best approximate the ideal frequency response while satisfying given design constraints

Window method

• The window method is a simple and efficient technique for designing FIR filters
• It involves multiplying an ideal impulse response (obtained from the desired frequency response) with a window function to obtain the actual filter coefficients
• The choice of the window function determines the trade-off between the transition band width and the stopband attenuation

Rectangular window

• The rectangular window is the simplest window function, with a constant value of 1 over the length of the impulse response
• FIR filters designed using the rectangular window have the narrowest main lobe width but exhibit the highest sidelobe levels, resulting in poor stopband attenuation

Hamming window

• The Hamming window is a popular window function that provides a good compromise between main lobe width and sidelobe levels
• It is defined as: $w[n] = 0.54 - 0.46 \cos\left(\frac{2\pi n}{N-1}\right)$, for $0 \leq n \leq N-1$
• FIR filters designed using the Hamming window have moderate transition bandwidth and stopband attenuation

Hann window

• The Hann window (also known as the Hanning window) is another commonly used window function
• It is defined as: $w[n] = 0.5 - 0.5 \cos\left(\frac{2\pi n}{N-1}\right)$, for $0 \leq n \leq N-1$
• FIR filters designed using the Hann window have slightly wider main lobe and lower sidelobe levels compared to the Hamming window

Blackman window

• The Blackman window is a window function that provides even lower sidelobe levels than the Hamming and Hann windows, at the cost of a wider main lobe
• It is defined as: $w[n] = 0.42 - 0.5 \cos\left(\frac{2\pi n}{N-1}\right) + 0.08 \cos\left(\frac{4\pi n}{N-1}\right)$, for $0 \leq n \leq N-1$
• FIR filters designed using the Blackman window have excellent stopband attenuation but a relatively wide transition band

Kaiser window

• The Kaiser window is a parametric window function that allows for a trade-off between the main lobe width and the sidelobe levels by adjusting a single parameter, $\beta$
• The Kaiser window is defined in terms of the zeroth-order modified Bessel function of the first kind, $I_0$
• Higher values of $\beta$ result in lower sidelobe levels but a wider main lobe, while lower values of $\beta$ give a narrower main lobe but higher sidelobe levels

Frequency sampling method

• The frequency sampling method is an FIR filter design technique that directly specifies the desired frequency response at a set of discrete frequency points
• The filter coefficients are then obtained by taking the inverse Fourier transform of the sampled frequency response
• This method allows for precise control over the frequency response at the sampled points but may result in ripples between the samples

Weighted least squares method

• The weighted least squares method is an FIR filter design technique that minimizes the weighted error between the desired and actual frequency responses
• The error is measured as the squared difference between the desired and actual responses, multiplied by a weighting function that emphasizes certain frequency regions
• This method allows for a trade-off between the approximation error in different frequency bands by adjusting the weighting function

Parks-McClellan algorithm

• The Parks-McClellan algorithm (also known as the equiripple or Remez exchange algorithm) is an iterative method for designing optimal FIR filters with equiripple error in the passband and stopband
• The algorithm aims to minimize the maximum weighted error between the desired and actual frequency responses, resulting in filters with optimal approximation properties
• FIR filters designed using the Parks-McClellan algorithm have equal ripple amplitudes in the passband and stopband, and the error is distributed evenly across the frequency bands of interest

FIR filter implementation

• FIR filters can be implemented efficiently in hardware or software using various structures and techniques
• The choice of implementation structure depends on factors such as the filter order, the available resources, and the desired performance characteristics

Direct form structure

• The direct form structure is the most straightforward way to implement an FIR filter
• It directly follows the difference equation, where the output is computed as a weighted sum of the current and past input samples
• The direct form structure consists of a set of delay elements, multipliers, and adders, with the filter coefficients stored in memory
• The number of multiplications and additions required per output sample is equal to the filter order, $N-1$

Transposed structure

• The transposed structure is an alternative implementation of FIR filters that has the same input-output relationship as the direct form structure
• It is obtained by reversing the signal flow graph of the direct form structure, exchanging the input and output, and replacing the delays with advances
• The transposed structure has the advantage of reduced latency compared to the direct form structure, as the input samples are immediately multiplied by the filter coefficients
• It also has better numerical properties, such as reduced round-off noise accumulation

Polyphase decomposition

• Polyphase decomposition is a technique for efficiently implementing FIR filters in multirate signal processing applications, such as interpolation and decimation
• The filter coefficients are divided into subsets (polyphase components) that operate on different phases of the input or output signal
• Polyphase decomposition reduces the computational complexity of multirate FIR filters by exploiting the redundancy in the computations
• It allows for efficient implementation of interpolation and decimation filters, as well as for the realization of fractional delay filters

Multirate techniques

• Multirate techniques involve changing the sampling rate of a signal by a factor of $L$ (interpolation) or $M$ (decimation)
• FIR filters are commonly used in multirate signal processing to prevent aliasing during decimation and to remove imaging during interpolation
• Interpolation by a factor of $L$ is achieved by inserting $L-1$ zeros between each input sample and then applying a lowpass filter to remove the imaging components
• Decimation by a factor of $M$ is achieved by first applying a lowpass filter to limit the bandwidth of the signal and then discarding $M-1$ out of every $M$ output samples
• Multirate FIR filters can be efficiently implemented using polyphase decomposition and noble identities, which allow for the reduction of the computational complexity

Comparison of FIR and IIR filters

• Finite Impulse Response (FIR) and Infinite Impulse Response (IIR) filters are two main types of digital filters used in signal processing
• Each type has its own advantages and disadvantages, and the choice between them depends on the specific application requirements

Advantages of FIR filters

• FIR filters can have an exact linear phase response, which is important for applications where the waveform shape must be preserved
• They are always stable, as they have no feedback and their impulse response is finite in duration
• FIR filters can be easily designed to have a desired frequency response using various methods, such as the window method or the Parks-McClellan algorithm
• They are less sensitive to quantization errors and round-off noise compared to IIR filters
• FIR filters are suitable for applications that require a constant group delay, such as audio and speech processing

Disadvantages of FIR filters

• FIR filters generally require a higher filter order (longer impulse response) than IIR filters to achieve a similar frequency response, leading to increased computational complexity and memory requirements
• The higher order of FIR filters also results in a longer group delay, which may be undesirable in certain real-time applications
• FIR filters are less efficient than IIR filters in terms of the number of arithmetic operations required per output sample
• The design of high-order FIR filters may be more challenging and time-consuming compared to IIR filters

Applications of FIR filters

• FIR filters find extensive use in various signal processing applications across different domains
• Their stability, linear phase response, and design flexibility make them suitable for a wide range of tasks

Audio signal processing

• FIR filters are commonly used in audio signal processing for tasks such as equalization, noise reduction, and echo cancellation
• They can be designed to have a desired frequency response, allowing for the selective enhancement or attenuation of specific frequency ranges
• Examples include graphic equalizers, parametric equalizers, and notch filters for removing unwanted frequencies

Image processing

• FIR filters are used in image processing for tasks such as smoothing, sharpening, and edge detection
• They can be designed as 2D filters, where the filter coefficients are arranged in a matrix to operate on the pixels of an image
• Examples include Gaussian smoothing filters, Laplacian filters for edge detection, and unsharp masking filters for image sharpening

Biomedical signal processing

• FIR filters are used in biomedical signal processing for the analysis and interpretation of physiological signals, such as electrocardiogram (ECG), electroencephalogram (EEG), and electromyogram (EMG)
• They are employed for tasks such as noise reduction, baseline wander removal, and feature extraction
• Examples include lowpass filters for removing high-frequency noise, highpass filters for eliminating baseline drift, and bandpass filters for isolating specific frequency components

Radar signal processing

• FIR filters are used in radar signal processing for tasks such as clutter suppression, range resolution enhancement, and Doppler processing
• They can be designed to have a desired impulse response or frequency response, depending on the specific requirements of the radar application
• Examples include matched filters for maximizing the signal-to-noise ratio, pulse compression filters for improving range resolution, and moving target indication (MTI) filters for suppressing stationary clutter