LTI_FiltersLinear Time-Invariant (LTI) filters are fundamental tools in signal processing that operate with consistent behavior over time and linearly respond to input signals. They are crucial for analyzing and manipulating signals in various applications, ensuring the output signal's integrity is maintained regardless of when an input is applied or its magnitude. The Windowed Sinc filter is a specific type of LTI filter designed for digital signal processing. It employs a Sinc function, ideal for low-pass filtering, truncated and shaped within a finite window to make it practically implementable. This process involves multiplying the Sinc function by a window function, which tapers off towards the ends, making the filter finite and suitable for digital applications. Windowed Sinc filters are particularly effective for tasks like data smoothing and removing unwanted frequency components, balancing between sharp cutoff characteristics and minimal distortion. The efficiency of Windowed Sinc filters in digital signal processing lies in their adept use of linear algebra, particularly in the convolution process, which combines input data with filter coefficients to produce the desired output. This mathematical foundation allows for precise control over the filtering process, optimizing the balance between filtering performance and computational efficiency. By leveraging linear algebra techniques such as matrix multiplication and Toeplitz matrices, these filters can efficiently handle large datasets and complex filtering tasks, making them invaluable in applications requiring high precision and speed, such as audio processing, financial signal analysis, and image restoration.
Library "LTI_Filters"
offset(length, enable)
Calculates the time offset required for aligning the output of a filter with its input, based on the filter's length. This is useful for centered filters where the output is naturally shifted due to the filter's operation.
Parameters:
length (simple int) : The length of the filter.
enable (simple bool) : A boolean flag to enable or dissable the offset calculation.
Returns: The calculated offset if enabled; otherwise, returns 0.
lti_filter(filter_type, source, length, prefilter, centered, fc, window_type)
General-purpose Linear Time-Invariant (LTI) filter function that can apply various filter types to a data series. Can be used to apply a variety of LTI filters with different characteristics to financial data series or other time series data.
Parameters:
filter_type (simple string) : Specifies the type of filter. ("Sinc", "SMA", "WMA")
source (float) : The input data series to filter.
length (simple int) : The length of the filter.
prefilter (simple bool) : Boolean indicating whether to prefilter the input data.
centered (simple bool) : Determines whether the filter coefficients are centered.
fc (simple float) : Filter cutoff. Expressed like a length.
window_type (simple string) : Type of window function to apply. ("Hann", "Hamming", "Blackman", "Triangular", "Lanczos", "None")
Returns: The filtered data series.
lti_sma(source, length, prefilter)
Applies a Simple Moving Average (SMA) filter to the data series. Useful for smoothing data series to identify trends or for use as a component in more complex indicators.
Parameters:
source (float) : The input data series to filter.
length (simple int) : The length of the SMA filter.
prefilter (simple bool) : Boolean indicating whether to prefilter the input data.
Returns: The SMA-filtered data series.
lti_wma(source, length, prefilter, centered)
Applies a Weighted Moving Average (WMA) filter to a data series. Ideal for smoothing data with emphasis on more recent values, allowing for dynamic adjustments to the weighting scheme.
Parameters:
source (float) : The input data series to filter.
length (simple int) : The length of the WMA filter.
prefilter (simple bool) : Boolean indicating whether to prefilter the input data.
centered (simple bool) : Determines whether the filter coefficients are centered.
Returns: The WMA-filtered data series.
lti_sinc(source, length, prefilter, centered, fc, window_type)
Applies a Sinc filter to a data series, optionally using a window function. Particularly useful for signal processing tasks within financial analysis, such as smoothing or trend identification, with the ability to fine-tune filter characteristics.
Parameters:
source (float) : The input data series to filter.
length (simple int) : The length of the Sinc filter.
prefilter (simple bool) : Boolean indicating whether to prefilter the input data.
centered (simple bool) : Determines whether the filter coefficients are centered.
fc (simple float) : Filter cutoff. Expressed like a length.
window_type (simple string) : Type of window function to apply. ("Hann", "Hamming", "Blackman", "Triangular", "Lanczos", "None")
Returns: The Sinc-filtered data series.
SINC
Filtered, N-Order Power-of-Cosine, Sinc FIR Filter [Loxx]Filtered, N-Order Power-of-Cosine, Sinc FIR Filter is a Discrete-Time, FIR Digital Filter that uses Power-of-Cosine Family of FIR filters. This is an N-order algorithm that allows up to 50 values for alpha, orders, of depth. This one differs from previous Power-of-Cosine filters I've published in that it this uses Windowed-Sinc filtering. I've also included a Dual Element Lag Reducer using Kalman velocity, a standard deviation filter, and a clutter filter. You can read about each of these below.
Impulse Response
What are FIR Filters?
In discrete-time signal processing, windowing is a preliminary signal shaping technique, usually applied to improve the appearance and usefulness of a subsequent Discrete Fourier Transform. Several window functions can be defined, based on a constant (rectangular window), B-splines, other polynomials, sinusoids, cosine-sums, adjustable, hybrid, and other types. The windowing operation consists of multipying the given sampled signal by the window function. For trading purposes, these FIR filters act as advanced weighted moving averages.
A finite impulse response (FIR) filter is a filter whose impulse response (or response to any finite length input) is of finite duration, because it settles to zero in finite time. This is in contrast to infinite impulse response (IIR) filters, which may have internal feedback and may continue to respond indefinitely (usually decaying).
The impulse response (that is, the output in response to a Kronecker delta input) of an Nth-order discrete-time FIR filter lasts exactly {\displaystyle N+1}N+1 samples (from first nonzero element through last nonzero element) before it then settles to zero.
FIR filters can be discrete-time or continuous-time, and digital or analog.
A FIR filter is (similar to, or) just a weighted moving average filter, where (unlike a typical equally weighted moving average filter) the weights of each delay tap are not constrained to be identical or even of the same sign. By changing various values in the array of weights (the impulse response, or time shifted and sampled version of the same), the frequency response of a FIR filter can be completely changed.
An FIR filter simply CONVOLVES the input time series (price data) with its IMPULSE RESPONSE. The impulse response is just a set of weights (or "coefficients") that multiply each data point. Then you just add up all the products and divide by the sum of the weights and that is it; e.g., for a 10-bar SMA you just add up 10 bars of price data (each multiplied by 1) and divide by 10. For a weighted-MA you add up the product of the price data with triangular-number weights and divide by the total weight.
What is a Standard Deviation Filter?
If price or output or both don't move more than the (standard deviation) * multiplier then the trend stays the previous bar trend. This will appear on the chart as "stepping" of the moving average line. This works similar to Super Trend or Parabolic SAR but is a more naive technique of filtering.
What is a Clutter Filter?
For our purposes here, this is a filter that compares the slope of the trading filter output to a threshold to determine whether to shift trends. If the slope is up but the slope doesn't exceed the threshold, then the color is gray and this indicates a chop zone. If the slope is down but the slope doesn't exceed the threshold, then the color is gray and this indicates a chop zone. Alternatively if either up or down slope exceeds the threshold then the trend turns green for up and red for down. Fro demonstration purposes, an EMA is used as the moving average. This acts to reduce the noise in the signal.
What is a Dual Element Lag Reducer?
Modifies an array of coefficients to reduce lag by the Lag Reduction Factor uses a generic version of a Kalman velocity component to accomplish this lag reduction is achieved by applying the following to the array:
2 * coeff - coeff
The response time vs noise battle still holds true, high lag reduction means more noise is present in your data! Please note that the beginning coefficients which the modifying matrix cannot be applied to (coef whose indecies are < LagReductionFactor) are simply multiplied by two for additional smoothing .
Whats a Windowed-Sinc Filter?
Windowed-sinc filters are used to separate one band of frequencies from another. They are very stable, produce few surprises, and can be pushed to incredible performance levels. These exceptional frequency domain characteristics are obtained at the expense of poor performance in the time domain, including excessive ripple and overshoot in the step response. When carried out by standard convolution, windowed-sinc filters are easy to program, but slow to execute.
The sinc function sinc (x), also called the "sampling function," is a function that arises frequently in signal processing and the theory of Fourier transforms.
In mathematics, the historical unnormalized sinc function is defined for x ≠ 0 by
sinc x = sinx / x
In digital signal processing and information theory, the normalized sinc function is commonly defined for x ≠ 0 by
sinc x = sin(pi * x) / (pi * x)
For our purposes here, we are used a normalized Sinc function
Included
Bar coloring
Loxx's Expanded Source Types
Signals
Alerts
Related indicators
Variety, Low-Pass, FIR Filter Impulse Response Explorer
STD-Filtered, Variety FIR Digital Filters w/ ATR Bands
STD/C-Filtered, N-Order Power-of-Cosine FIR Filter
STD/C-Filtered, Truncated Taylor Family FIR Filter
STD/Clutter-Filtered, Kaiser Window FIR Digital Filter
STD/Clutter Filtered, One-Sided, N-Sinc-Kernel, EFIR Filt
Variety, Low-Pass, FIR Filter Impulse Response Explorer [Loxx]Variety Low-Pass FIR Filter, Impulse Response Explorer is a simple impulse response explorer of 16 of the most popular FIR digital filtering windowing techniques. Y-values are the values of the coefficients produced by the selected algorithms; X-values are the index of sample. This indicator also allows you to turn on Sinc Windowing for all window types except for Rectangular, Triangular, and Linear. This is an educational indicator to demonstrate the differences between popular FIR filters in terms of their coefficient outputs. This is also used to compliment other indicators I've published or will publish that implement advanced FIR digital filters (see below to find applicable indicators).
Inputs:
Number of Coefficients to Calculate = Sample size; for example, this would be the period used in SMA or WMA
FIR Digital Filter Type = FIR windowing method you would like to explore
Multiplier (Sinc only) = applies a multiplier effect to the Sinc Windowing
Frequency Cutoff = this is necessary to smooth the output and get rid of noise. the lower the number, the smoother the output.
Turn on Sinc? = turn this on if you want to convert the windowing function from regular function to a Windowed-Sinc filter
Order = This is used for power of cosine filter only. This is the N-order, or depth, of the filter you wish to create.
What are FIR Filters?
In discrete-time signal processing, windowing is a preliminary signal shaping technique, usually applied to improve the appearance and usefulness of a subsequent Discrete Fourier Transform. Several window functions can be defined, based on a constant (rectangular window), B-splines, other polynomials, sinusoids, cosine-sums, adjustable, hybrid, and other types. The windowing operation consists of multipying the given sampled signal by the window function. For trading purposes, these FIR filters act as advanced weighted moving averages.
A finite impulse response (FIR) filter is a filter whose impulse response (or response to any finite length input) is of finite duration, because it settles to zero in finite time. This is in contrast to infinite impulse response (IIR) filters, which may have internal feedback and may continue to respond indefinitely (usually decaying).
The impulse response (that is, the output in response to a Kronecker delta input) of an Nth-order discrete-time FIR filter lasts exactly {\displaystyle N+1}N+1 samples (from first nonzero element through last nonzero element) before it then settles to zero.
FIR filters can be discrete-time or continuous-time, and digital or analog.
A FIR filter is (similar to, or) just a weighted moving average filter, where (unlike a typical equally weighted moving average filter) the weights of each delay tap are not constrained to be identical or even of the same sign. By changing various values in the array of weights (the impulse response, or time shifted and sampled version of the same), the frequency response of a FIR filter can be completely changed.
An FIR filter simply CONVOLVES the input time series (price data) with its IMPULSE RESPONSE. The impulse response is just a set of weights (or "coefficients") that multiply each data point. Then you just add up all the products and divide by the sum of the weights and that is it; e.g., for a 10-bar SMA you just add up 10 bars of price data (each multiplied by 1) and divide by 10. For a weighted-MA you add up the product of the price data with triangular-number weights and divide by the total weight.
What's a Low-Pass Filter?
A low-pass filter is the type of frequency domain filter that is used for smoothing sound, image, or data. This is different from a high-pass filter that is used for sharpening data, images, or sound.
Whats a Windowed-Sinc Filter?
Windowed-sinc filters are used to separate one band of frequencies from another. They are very stable, produce few surprises, and can be pushed to incredible performance levels. These exceptional frequency domain characteristics are obtained at the expense of poor performance in the time domain, including excessive ripple and overshoot in the step response. When carried out by standard convolution, windowed-sinc filters are easy to program, but slow to execute.
The sinc function sinc (x), also called the "sampling function," is a function that arises frequently in signal processing and the theory of Fourier transforms.
In mathematics, the historical unnormalized sinc function is defined for x ≠ 0 by
sinc x = sinx / x
In digital signal processing and information theory, the normalized sinc function is commonly defined for x ≠ 0 by
sinc x = sin(pi * x) / (pi * x)
For our purposes here, we are used a normalized Sinc function
Included Windowing Functions
N-Order Power-of-Cosine (this one is really N-different types of FIR filters)
Hamming
Hanning
Blackman
Blackman Harris
Blackman Nutall
Nutall
Bartlet Zero End Points
Bartlet-Hann
Hann
Sine
Lanczos
Flat Top
Rectangular
Linear
Triangular
If you wish to dive deeper to get a full explanation of these windowing functions, see here: en.wikipedia.org
Related indicators
STD-Filtered, Variety FIR Digital Filters w/ ATR Bands
STD/C-Filtered, N-Order Power-of-Cosine FIR Filter
STD/C-Filtered, Truncated Taylor Family FIR Filter
STD/Clutter-Filtered, Kaiser Window FIR Digital Filter
STD/Clutter Filtered, One-Sided, N-Sinc-Kernel, EFIR Filt
STD/Clutter Filtered, One-Sided, N-Sinc-Kernel, EFIR Filt [Loxx]STD/Clutter Filtered, One-Sided, N-Sinc-Kernel, EFIR Filt is a normalized Cardinal Sine Filter Kernel Weighted Fir Filter that uses Ehler's FIR filter calculation instead of the general FIR filter calculation. This indicator has Kalman Velocity lag reduction, a standard deviation filter, a clutter filter, and a kernel noise filter. When calculating the Kernels, the both sides are calculated, then smoothed, then sliced to just the Right side of the Kernel weights. Lastly, blackman windowing is used for our purposes here. You can read about blackman windowing here:
Blackman window
Advantages of Blackman Window over Hamming Window Method for designing FIR Filter
The Kernel amplitudes are shown below with their corresponding values in yellow:
This indicator is intended to be used with Heikin-Ashi source inputs, specially HAB Median. You can read about this here:
Moving Average Filters Add-on w/ Expanded Source Types
What is a Finite Impulse Response Filter?
In signal processing, a finite impulse response (FIR) filter is a filter whose impulse response (or response to any finite length input) is of finite duration, because it settles to zero in finite time. This is in contrast to infinite impulse response (IIR) filters, which may have internal feedback and may continue to respond indefinitely (usually decaying).
The impulse response (that is, the output in response to a Kronecker delta input) of an Nth-order discrete-time FIR filter lasts exactly {\displaystyle N+1}N+1 samples (from first nonzero element through last nonzero element) before it then settles to zero.
FIR filters can be discrete-time or continuous-time, and digital or analog.
A FIR filter is (similar to, or) just a weighted moving average filter, where (unlike a typical equally weighted moving average filter) the weights of each delay tap are not constrained to be identical or even of the same sign. By changing various values in the array of weights (the impulse response, or time shifted and sampled version of the same), the frequency response of a FIR filter can be completely changed.
An FIR filter simply CONVOLVES the input time series (price data) with its IMPULSE RESPONSE. The impulse response is just a set of weights (or "coefficients") that multiply each data point. Then you just add up all the products and divide by the sum of the weights and that is it; e.g., for a 10-bar SMA you just add up 10 bars of price data (each multiplied by 1) and divide by 10. For a weighted-MA you add up the product of the price data with triangular-number weights and divide by the total weight.
Ultra Low Lag Moving Average's weights are designed to have MAXIMUM possible smoothing and MINIMUM possible lag compatible with as-flat-as-possible phase response.
Ehlers FIR Filter
Ehlers Filter (EF) was authored, not surprisingly, by John Ehlers. Read all about them here: Ehlers Filters
What is Normalized Cardinal Sine?
The sinc function sinc (x), also called the "sampling function," is a function that arises frequently in signal processing and the theory of Fourier transforms.
In mathematics, the historical unnormalized sinc function is defined for x ≠ 0 by
sinc x = sinx / x
In digital signal processing and information theory, the normalized sinc function is commonly defined for x ≠ 0 by
sinc x = sin(pi * x) / (pi * x)
What is a Clutter Filter?
For our purposes here, this is a filter that compares the slope of the trading filter output to a threshold to determine whether to shift trends. If the slope is up but the slope doesn't exceed the threshold, then the color is gray and this indicates a chop zone. If the slope is down but the slope doesn't exceed the threshold, then the color is gray and this indicates a chop zone. Alternatively if either up or down slope exceeds the threshold then the trend turns green for up and red for down. Fro demonstration purposes, an EMA is used as the moving average. This acts to reduce the noise in the signal.
What is a Dual Element Lag Reducer?
Modifies an array of coefficients to reduce lag by the Lag Reduction Factor uses a generic version of a Kalman velocity component to accomplish this lag reduction is achieved by applying the following to the array:
2 * coeff - coeff
The response time vs noise battle still holds true, high lag reduction means more noise is present in your data! Please note that the beginning coefficients which the modifying matrix cannot be applied to (coef whose indecies are < LagReductionFactor) are simply multiplied by two for additional smoothing .
Included
Bar coloring
Loxx's Expanded Source Types
Signals
Alerts
STD-Filtered, Ultra Low Lag Moving Average [Loxx]STD-Filtered, Ultra Low Lag Moving Average is a FIR filter that smooths price using a low-pass filtering with weights derived from a normalized cardinal since function. This indicator attempts to reduce lag to an extreme degree. Try this on various time frames with various Type inputs, 0 is the default, so see where the sweet spot is for your trading style.
What is a Finite Impulse Response Filter?
In signal processing, a finite impulse response (FIR) filter is a filter whose impulse response (or response to any finite length input) is of finite duration, because it settles to zero in finite time. This is in contrast to infinite impulse response (IIR) filters, which may have internal feedback and may continue to respond indefinitely (usually decaying).
The impulse response (that is, the output in response to a Kronecker delta input) of an Nth-order discrete-time FIR filter lasts exactly {\displaystyle N+1}N+1 samples (from first nonzero element through last nonzero element) before it then settles to zero.
FIR filters can be discrete-time or continuous-time, and digital or analog.
A FIR filter is (similar to, or) just a weighted moving average filter, where (unlike a typical equally weighted moving average filter) the weights of each delay tap are not constrained to be identical or even of the same sign. By changing various values in the array of weights (the impulse response, or time shifted and sampled version of the same), the frequency response of a FIR filter can be completely changed.
An FIR filter simply CONVOLVES the input time series (price data) with its IMPULSE RESPONSE. The impulse response is just a set of weights (or "coefficients") that multiply each data point. Then you just add up all the products and divide by the sum of the weights and that is it; e.g., for a 10-bar SMA you just add up 10 bars of price data (each multiplied by 1) and divide by 10. For a weighted-MA you add up the product of the price data with triangular-number weights and divide by the total weight.
Ultra Low Lag Moving Average's weights are designed to have MAXIMUM possible smoothing and MINIMUM possible lag compatible with as-flat-as-possible phase response.
What is Normalized Cardinal Sine?
The sinc function sinc(x), also called the "sampling function," is a function that arises frequently in signal processing and the theory of Fourier transforms.
In mathematics, the historical unnormalized sinc function is defined for x ≠ 0 by
sinc x = sinx / x
In digital signal processing and information theory, the normalized sinc function is commonly defined for x ≠ 0 by
sinc x = sin(pi * x) / (pi * x)
How this works, (easy mode)
1. Use a HA or HAB source type
2. The lower the Type value the smoother the moving average
3. Standard deviation stepping is added to further reduce noise
Included
Bar coloring
Signals
Alerts
Loxx's Expanded Source Types
