|
|
|
@@ -27,7 +27,7 @@ Image credits are from Wikipedia. |
|
|
|
|
|
|
|
### Signals |
|
|
|
|
|
|
|
A *signal* is a function $x(t): \mathbb{R} \rightarrow \mathbb{R}$ of amplitudes (voltages, sound pressure levels, etc.) defined on a time continuum, and a *sequence* is a function $x(n): \mathbb{Z} \rightarrow \mathbb{R}$ defined only at integer points, often written as $x_n$. |
|
|
|
A *signal* is a function \\(x(t): \mathbb{R} \rightarrow \mathbb{R}\\) of amplitudes (voltages, sound pressure levels, etc.) defined on a time continuum, and a *sequence* is a function \\(x(n): \mathbb{Z} \rightarrow \mathbb{R}\\) defined only at integer points, often written as \\(x_n\\). |
|
|
|
In other words, a signal is an infinitely long continuum of values with infinite time detail, and a sequence is an infinitely long stream of samples at evenly-spaced isolated points in time. |
|
|
|
|
|
|
|
Analog hardware produces and processes signals, while digital algorithms handle sequences of samples (often called *digital signals*.) |
|
|
|
@@ -35,31 +35,31 @@ Analog hardware produces and processes signals, while digital algorithms handle |
|
|
|
|
|
|
|
### Fourier analysis |
|
|
|
|
|
|
|
In DSP you often encounter periodic signals that repeat at a frequency $f$, like cyclic waveforms. |
|
|
|
You may be aware that periodic signals are composed of multiple harmonics (shifted cosine waves) with frequencies $fk$ for $k = 1, 2, 3, \ldots$. |
|
|
|
In DSP you often encounter periodic signals that repeat at a frequency \\(f\\), like cyclic waveforms. |
|
|
|
You may be aware that periodic signals are composed of multiple harmonics (shifted cosine waves) with frequencies \\(fk\\) for \\(k = 1, 2, 3, \ldots\\). |
|
|
|
In fact, a periodic signal can be *defined* by the amplitudes and phases of all of its harmonics. |
|
|
|
For simplicity, if $x(t)$ is a periodic signal with $f=1$, then |
|
|
|
$$ x(t) = A_0 + \sum_{k=1}^\infty A_k \cos \left( 2\pi k t + \phi_k \right) $$ |
|
|
|
where $A_0$ is the *DC component* (the signal's average offset from zero), $A_k$ is the amplitude of harmonic $k$, and $\phi_k$ is its angular phase. |
|
|
|
For simplicity, if \\(x(t)\\) is a periodic signal with \\(f=1\\), then |
|
|
|
\\[ x(t) = A_0 + \sum_{k=1}^\infty A_k \cos \left( 2\pi k t + \phi_k \right) \\] |
|
|
|
where \\(A_0\\) is the *DC component* (the signal's average offset from zero), \\(A_k\\) is the amplitude of harmonic \\(k\\), and \\(\phi_k\\) is its angular phase. |
|
|
|
Each term of the sum is a simple harmonic signal, and when superimposed, they reconstruct the original signal. |
|
|
|
|
|
|
|
As it turns out, not only periodic signals but *all* signals have a unique decomposition into shifted cosines. |
|
|
|
However, instead of integer harmonics, you must consider all frequencies $f$, and instead of a discrete sum, you must integrate over all shifted cosines of real-numbered frequencies. |
|
|
|
Furthermore, it is easier to combine the amplitude and phase into a single complex-valued component $X(f) = A(f) e^{i \phi(f)}$. |
|
|
|
However, instead of integer harmonics, you must consider all frequencies \\(f\\), and instead of a discrete sum, you must integrate over all shifted cosines of real-numbered frequencies. |
|
|
|
Furthermore, it is easier to combine the amplitude and phase into a single complex-valued component \\(X(f) = A(f) e^{i \phi(f)}\\). |
|
|
|
With these modifications, the decomposition becomes |
|
|
|
$$ x(t) = \int_{-\infty}^\infty X(f) e^{2\pi i t f} df $$ |
|
|
|
The $X(f)$ component can be calculated with |
|
|
|
$$ X(f) = \int_{-\infty}^\infty x(t) e^{-2\pi i t f} dt $$ |
|
|
|
\\[ x(t) = \int_{-\infty}^\infty X(f) e^{2\pi i t f} df \\] |
|
|
|
The \\(X(f)\\) component can be calculated with |
|
|
|
\\[ X(f) = \int_{-\infty}^\infty x(t) e^{-2\pi i t f} dt \\] |
|
|
|
This is known as the [Fourier transform](https://en.wikipedia.org/wiki/Fourier_transform). |
|
|
|
|
|
|
|
For a digital periodic signal with period $N$, no harmonics above $N/2$ can be represented, so its decomposition is truncated. |
|
|
|
$$ x_k = \sum_{n=0}^{N/2} X_n e^{2\pi i kn / N} $$ |
|
|
|
For a digital periodic signal with period \\(N\\), no harmonics above \\(N/2\\) can be represented, so its decomposition is truncated. |
|
|
|
\\[ x_k = \sum_{n=0}^{N/2} X_n e^{2\pi i kn / N} \\] |
|
|
|
Its inverse is |
|
|
|
$$ X_n = \sum_{k=0}^{N} x_k e^{-2\pi i kn / N} $$ |
|
|
|
Since the sequence $x_k$ is real-valued, the components $X_n$ are conjugate symmetric across $N/2$, i.e. $X_n = X_{N-n}^\ast$, so we have $N/2 + 1$ unique harmonic components. |
|
|
|
\\[ X_n = \sum_{k=0}^{N} x_k e^{-2\pi i kn / N} \\] |
|
|
|
Since the sequence \\(x_k\\) is real-valued, the components \\(X_n\\) are conjugate symmetric across \\(N/2\\), i.e. \\(X_n = X_{N-n}^\ast\\), so we have \\(N/2 + 1\\) unique harmonic components. |
|
|
|
This is the (real) [discrete Fourier transform](https://en.wikipedia.org/wiki/Discrete_Fourier_transform) (DFT). |
|
|
|
|
|
|
|
The [fast Fourier transform](https://en.wikipedia.org/wiki/Fast_Fourier_transform) (FFT) is a method that surprisingly computes the DFT of a block of $N$ samples in $O(N \log N)$ steps rather than $O(N^2)$ like the DFT by using a recursive [dynamic programming](https://en.wikipedia.org/wiki/Dynamic_programming) algorithm, while being more numerically stable than the naive sum. |
|
|
|
The [fast Fourier transform](https://en.wikipedia.org/wiki/Fast_Fourier_transform) (FFT) is a method that surprisingly computes the DFT of a block of \\(N\\) samples in \\(O(N \log N)\\) steps rather than \\(O(N^2)\\) like the DFT by using a recursive [dynamic programming](https://en.wikipedia.org/wiki/Dynamic_programming) algorithm, while being more numerically stable than the naive sum. |
|
|
|
This makes the DFT feasible in high-performance DSP when processing blocks of samples. |
|
|
|
The larger the block size, the lower the average number of steps required per sample. |
|
|
|
FFT algorithms exist in many libraries, like [pffft](https://bitbucket.org/jpommier/pffft/) used in VCV Rack itself. |
|
|
|
@@ -67,11 +67,11 @@ FFT algorithms exist in many libraries, like [pffft](https://bitbucket.org/jpomm |
|
|
|
|
|
|
|
### Sampling |
|
|
|
|
|
|
|
The [Nyquist–Shannon sampling theorem](https://en.wikipedia.org/wiki/Nyquist%E2%80%93Shannon_sampling_theorem) states that a signal with no frequency components higher than half the sample rate $f_{sr}$ can be sampled and reconstructed without losing information. |
|
|
|
In other words, if you bandlimit a signal (with a brickwall lowpass filter at $f_{sr}/2$) and sample points at $f_{sr}$, you can reconstruct the bandlimited signal by finding the unique signal which passes through all points and has no frequency components higher than $f_{sr}/2$. |
|
|
|
The [Nyquist–Shannon sampling theorem](https://en.wikipedia.org/wiki/Nyquist%E2%80%93Shannon_sampling_theorem) states that a signal with no frequency components higher than half the sample rate \\(f_{sr}\\) can be sampled and reconstructed without losing information. |
|
|
|
In other words, if you bandlimit a signal (with a brickwall lowpass filter at \\(f_{sr}/2\\)) and sample points at \\(f_{sr}\\), you can reconstruct the bandlimited signal by finding the unique signal which passes through all points and has no frequency components higher than \\(f_{sr}/2\\). |
|
|
|
|
|
|
|
In practice, digital-to-analog converters (DACs) apply an approximation of a brickwall lowpass filter to remove frequencies higher than $f_{sr}/2$ from the signal. |
|
|
|
The signal is integrated for a small fraction of the sample time $1/f_{sr}$ to obtain an approximation of the amplitude at a point in time, and this measurement is quantized to the nearest digital value. |
|
|
|
In practice, digital-to-analog converters (DACs) apply an approximation of a brickwall lowpass filter to remove frequencies higher than \\(f_{sr}/2\\) from the signal. |
|
|
|
The signal is integrated for a small fraction of the sample time \\(1/f_{sr}\\) to obtain an approximation of the amplitude at a point in time, and this measurement is quantized to the nearest digital value. |
|
|
|
|
|
|
|
Analog-to-digital converters (ADCs) convert a digital value to an amplitude and hold it for a fraction of the sample time. |
|
|
|
A [reconstruction filter](https://en.wikipedia.org/wiki/Reconstruction_filter) is applied, producing a signal close to the original bandlimited signal. |
|
|
|
@@ -84,7 +84,7 @@ Fortunately, modern DACs and ADCs as cheap as $2-5 per chip can digitize and |
|
|
|
|
|
|
|
### Aliasing |
|
|
|
|
|
|
|
The Nyquist–Shannon sampling theorem requires the original signal to be bandlimited at $f_{sr}/2$ before digitizing. |
|
|
|
The Nyquist–Shannon sampling theorem requires the original signal to be bandlimited at \\(f_{sr}/2\\) before digitizing. |
|
|
|
If it is not, reconstructing will result in an entirely different signal, which usually sounds ugly and is associated with poor-quality DSP. |
|
|
|
|
|
|
|
Consider the high-frequency sine wave in red. |
|
|
|
@@ -93,16 +93,16 @@ If correctly bandlimited, the original signal would be zero (silence), and thus |
|
|
|
|
|
|
|
[](https://en.wikipedia.org/wiki/File:AliasingSines.svg) |
|
|
|
|
|
|
|
A square wave has harmonic amplitudes $\frac{1}{k}$ for odd harmonics $k$. |
|
|
|
However, after bandlimiting, all harmonics above $f_{sr}/2$ become zero, so its reconstruction should look like this. |
|
|
|
A square wave has harmonic amplitudes \\(\frac{1}{k}\\) for odd harmonics \\(k\\). |
|
|
|
However, after bandlimiting, all harmonics above \\(f_{sr}/2\\) become zero, so its reconstruction should look like this. |
|
|
|
|
|
|
|
[](https://en.wikipedia.org/wiki/File:Gibbs_phenomenon_50.svg) |
|
|
|
|
|
|
|
The curve produced by a bandlimited discontinuity is known as the [Gibbs phenomenon](https://en.wikipedia.org/wiki/Gibbs_phenomenon). |
|
|
|
A DSP algorithm attempting to model a jump found in sawtooth or square waves must include this effect, such as by inserting a minBLEP or polyBLEP signal for each discontinuity. |
|
|
|
Otherwise, higher harmonics, like the high-frequency sine wave above, will pollute the spectrum below $f_{sr}/2$. |
|
|
|
Otherwise, higher harmonics, like the high-frequency sine wave above, will pollute the spectrum below \\(f_{sr}/2\\). |
|
|
|
|
|
|
|
Even signals containing no discontinuities, such as a triangle wave with harmonic amplitudes $(-1)^k / k^2$, must be correctly bandlimited or aliasing will occur. |
|
|
|
Even signals containing no discontinuities, such as a triangle wave with harmonic amplitudes \\((-1)^k / k^2\\), must be correctly bandlimited or aliasing will occur. |
|
|
|
One possible method is to realize that a triangle wave is an integrated square wave, and an integrator is just a filter with a -20dB per [decade](https://en.wikipedia.org/wiki/Decade_%28log_scale%29) slope. |
|
|
|
Since linear filters commute, a bandlimited integrated square wave is just an integrated bandlimited square wave. |
|
|
|
|
|
|
|
@@ -116,89 +116,89 @@ It is sometimes *not* required for reverb, linear filters, audio-rate FM of sine |
|
|
|
### Linear filters |
|
|
|
|
|
|
|
A linear filter is a operation that applies gain depending on a signal's frequency content, defined by |
|
|
|
$$Y(s) = H(s) X(s)$$ |
|
|
|
where $s = i \omega$ is the complex angular frequency, $X$ and $Y$ are the [Laplace transforms](https://en.wikipedia.org/wiki/Laplace_transform) of the input signal $x(t)$ and output signal $y(t)$, and $H(s)$ is the [transfer function](https://en.wikipedia.org/wiki/Transfer_function) of the filter, defining its character. |
|
|
|
\\[Y(s) = H(s) X(s)\\] |
|
|
|
where \\(s = i \omega\\) is the complex angular frequency, \\(X\\) and \\(Y\\) are the [Laplace transforms](https://en.wikipedia.org/wiki/Laplace_transform) of the input signal \\(x(t)\\) and output signal \\(y(t)\\), and \\(H(s)\\) is the [transfer function](https://en.wikipedia.org/wiki/Transfer_function) of the filter, defining its character. |
|
|
|
Note that the [Fourier transform](https://en.wikipedia.org/wiki/Fourier_transform) is not used because of time causality, i.e. we do not know the future of a signal. |
|
|
|
The filter is "linear" because the filtered sum of two signals is equal to the sum of the two individually filtered signals. |
|
|
|
|
|
|
|
A log-log plot of $H(i \omega)$ is called a [Bode plot](https://en.wikipedia.org/wiki/Bode_plot). |
|
|
|
A log-log plot of \\(H(i \omega)\\) is called a [Bode plot](https://en.wikipedia.org/wiki/Bode_plot). |
|
|
|
|
|
|
|
[](https://en.wikipedia.org/wiki/File:Butterworth_response.svg) |
|
|
|
|
|
|
|
To be able to exploit various mathematical tools, the transfer function is often written as a rational function in terms of $s^{-1}$ |
|
|
|
$$H(s) = \frac{\sum_{p=0}^P b_p s^{-p}}{\sum_{q=0}^Q a_q s^{-q}}$$ |
|
|
|
where $a_q$ and $b_p$ are called the *analog filter coefficients*. |
|
|
|
With sufficient orders $P$ and $Q$ of the numerator and denominator polynomial, you can approximate most linear analog filters found in synthesis. |
|
|
|
To be able to exploit various mathematical tools, the transfer function is often written as a rational function in terms of \\(s^{-1}\\) |
|
|
|
\\[H(s) = \frac{\sum_{p=0}^P b_p s^{-p}}{\sum_{q=0}^Q a_q s^{-q}}\\] |
|
|
|
where \\(a_q\\) and \\(b_p\\) are called the *analog filter coefficients*. |
|
|
|
With sufficient orders \\(P\\) and \\(Q\\) of the numerator and denominator polynomial, you can approximate most linear analog filters found in synthesis. |
|
|
|
|
|
|
|
To digitally implement a transfer function, define $z$ as the operator that transforms a sample $x_n$ to its following sample, i.e. $x_{n+1} = z[x_n]$. |
|
|
|
We can actually write this as a variable in terms of $s$ and the sample time $T = 1/f_{sr}$. |
|
|
|
(Consider a sine wave with angular frequency $\omega$. The $z$ operator shifts its phase as if we delayed by $T$ time.) |
|
|
|
$$z = e^{sT}$$ |
|
|
|
To digitally implement a transfer function, define \\(z\\) as the operator that transforms a sample \\(x_n\\) to its following sample, i.e. \\(x_{n+1} = z[x_n]\\). |
|
|
|
We can actually write this as a variable in terms of \\(s\\) and the sample time \\(T = 1/f_{sr}\\). |
|
|
|
(Consider a sine wave with angular frequency \\(\omega\\). The \\(z\\) operator shifts its phase as if we delayed by \\(T\\) time.) |
|
|
|
\\[z = e^{sT}\\] |
|
|
|
|
|
|
|
A first order approximation of this is |
|
|
|
$$z = \frac{e^{sT/2}}{e^{-sT/2}} \approx \frac{1 + sT/2}{1 - sT/2}$$ |
|
|
|
\\[z = \frac{e^{sT/2}}{e^{-sT/2}} \approx \frac{1 + sT/2}{1 - sT/2}\\] |
|
|
|
and its inverse is |
|
|
|
$$s = \frac{1}{T} \ln{z} \approx \frac{2}{T} \frac{1 - z^{-1}}{1 + z^{-1}}$$ |
|
|
|
\\[s = \frac{1}{T} \ln{z} \approx \frac{2}{T} \frac{1 - z^{-1}}{1 + z^{-1}}\\] |
|
|
|
|
|
|
|
This is known as the [Bilinear transform](https://en.wikipedia.org/wiki/Bilinear_transform). |
|
|
|
In digital form, the rational transfer function is written as |
|
|
|
$$H(z) = \frac{\sum_{n=0}^N b_n z^{-n}}{\sum_{m=0}^M a_m z^{-m}}$$ |
|
|
|
Note that the orders $N$ and $M$ are not necessarily equal to the orders $P$ and $Q$ of the analog form, and we obtain a new set of numbers $a_m$ and $b_n$ called the *digital filter coefficients*. |
|
|
|
\\[H(z) = \frac{\sum_{n=0}^N b_n z^{-n}}{\sum_{m=0}^M a_m z^{-m}}\\] |
|
|
|
Note that the orders \\(N\\) and \\(M\\) are not necessarily equal to the orders \\(P\\) and \\(Q\\) of the analog form, and we obtain a new set of numbers \\(a_m\\) and \\(b_n\\) called the *digital filter coefficients*. |
|
|
|
|
|
|
|
The *zeros* of the filter are the nonzero values of $z$ which give a zero numerator, and the *poles* are the nonzero values of $z$ which give a zero denominator. |
|
|
|
A linear filter is stable (its [impulse response](https://en.wikipedia.org/wiki/Impulse_response) converges to 0) if and only if all poles lie strictly within the complex unit circle, i.e. $|z| < 1$. |
|
|
|
The *zeros* of the filter are the nonzero values of \\(z\\) which give a zero numerator, and the *poles* are the nonzero values of \\(z\\) which give a zero denominator. |
|
|
|
A linear filter is stable (its [impulse response](https://en.wikipedia.org/wiki/Impulse_response) converges to 0) if and only if all poles lie strictly within the complex unit circle, i.e. \\(|z| < 1\\). |
|
|
|
|
|
|
|
We should now have all the tools we need to digitally implement any linear analog filter response $H(s)$ and vise-versa. |
|
|
|
We should now have all the tools we need to digitally implement any linear analog filter response \\(H(s)\\) and vise-versa. |
|
|
|
|
|
|
|
#### IIR filters |
|
|
|
|
|
|
|
An infinite impulse response (IIR) filter is a digital filter that implements all possible rational transfer functions. |
|
|
|
By multiplying the denominator of the rational $H(z)$ definition above on both sides and applying it to an input $x_k$ and output $y_k$, we obtain the difference relation |
|
|
|
$$\sum_{m=0}^M a_m y_{k-m} = \sum_{n=0}^N b_n x_{k-n}$$ |
|
|
|
Usually $a_0$ is normalized to 1, and $y_k$ can be written explicitly. |
|
|
|
$$y_k = \sum_{n=0}^N b_n x_{k-n} - \sum_{m=1}^M a_m y_{k-m}$$ |
|
|
|
By multiplying the denominator of the rational \\(H(z)\\) definition above on both sides and applying it to an input \\(x_k\\) and output \\(y_k\\), we obtain the difference relation |
|
|
|
\\[\sum_{m=0}^M a_m y_{k-m} = \sum_{n=0}^N b_n x_{k-n}\\] |
|
|
|
Usually \\(a_0\\) is normalized to 1, and \\(y_k\\) can be written explicitly. |
|
|
|
\\[y_k = \sum_{n=0}^N b_n x_{k-n} - \sum_{m=1}^M a_m y_{k-m}\\] |
|
|
|
This iterative process can be computed for each sample |
|
|
|
|
|
|
|
For $N, M = 2$, this is a [biquad filter](https://en.wikipedia.org/wiki/Digital_biquad_filter), which is very fast, numerically stable (assuming the transfer function itself is mathematical stable), and a reasonably good sounding filter. |
|
|
|
For \\(N, M = 2\\), this is a [biquad filter](https://en.wikipedia.org/wiki/Digital_biquad_filter), which is very fast, numerically stable (assuming the transfer function itself is mathematical stable), and a reasonably good sounding filter. |
|
|
|
|
|
|
|
$$H(z) = \frac{b_0 + b_1 z^{-1} + b_2 z^{-2}}{1 + a_1 z^{-1} + a_2 z^{-2}}$$ |
|
|
|
\\[H(z) = \frac{b_0 + b_1 z^{-1} + b_2 z^{-2}}{1 + a_1 z^{-1} + a_2 z^{-2}}\\] |
|
|
|
|
|
|
|
#### FIR filters |
|
|
|
|
|
|
|
A finite impulse response (FIR) filter is a specific case of an IIR filter with $M = 0$ (a transfer function denominator of 1). For an input and output signal, the difference relation is |
|
|
|
$$y_k = \sum_{n=0}^N b_n x_{k-n}$$ |
|
|
|
A finite impulse response (FIR) filter is a specific case of an IIR filter with \\(M = 0\\) (a transfer function denominator of 1). For an input and output signal, the difference relation is |
|
|
|
\\[y_k = \sum_{n=0}^N b_n x_{k-n}\\] |
|
|
|
They are computationally straightforward and always stable since they have no poles. |
|
|
|
|
|
|
|
Long FIR filters ($N \geq 128$) like FIR reverbs (yes, they are just linear filters) can be optimized through FFTs. |
|
|
|
Note that the above formula is the convolution between vectors $y$ and $b$, and by the [convolution theorem](https://en.wikipedia.org/wiki/Convolution_theorem), |
|
|
|
$$y \ast b = \mathcal{F}^{-1} \{ \mathcal{F}\{y\} \cdot \mathcal{F}\{b\} \}$$ |
|
|
|
where $\cdot$ is element-wise multiplication. |
|
|
|
Long FIR filters (\\(N \geq 128\\)) like FIR reverbs (yes, they are just linear filters) can be optimized through FFTs. |
|
|
|
Note that the above formula is the convolution between vectors \\(y\\) and \\(b\\), and by the [convolution theorem](https://en.wikipedia.org/wiki/Convolution_theorem), |
|
|
|
\\[y \ast b = \mathcal{F}^{-1} \{ \mathcal{F}\{y\} \cdot \mathcal{F}\{b\} \}\\] |
|
|
|
where \\(\cdot\\) is element-wise multiplication. |
|
|
|
|
|
|
|
While the naive FIR formula above requires $O(N)$ computations per sample, the FFT FIR method requires an average of $O(\log N)$ computations per sample when processing blocks of $N$ samples. |
|
|
|
A disadvantage of the FFT FIR method is that the signal must be delayed by $N$ samples before any output can be determined. |
|
|
|
You can combine the naive and FFT methods into a hybrid approach with the [overlap-add](https://en.wikipedia.org/wiki/Overlap%E2%80%93add_method) or [overlap-save](https://en.wikipedia.org/wiki/Overlap%E2%80%93save_method) methods, allowing you to process smaller blocks of size $M < N$ at a slightly worse $O((N/M) \log M)$ average computations per sample. |
|
|
|
While the naive FIR formula above requires \\(O(N)\\) computations per sample, the FFT FIR method requires an average of \\(O(\log N)\\) computations per sample when processing blocks of \\(N\\) samples. |
|
|
|
A disadvantage of the FFT FIR method is that the signal must be delayed by \\(N\\) samples before any output can be determined. |
|
|
|
You can combine the naive and FFT methods into a hybrid approach with the [overlap-add](https://en.wikipedia.org/wiki/Overlap%E2%80%93add_method) or [overlap-save](https://en.wikipedia.org/wiki/Overlap%E2%80%93save_method) methods, allowing you to process smaller blocks of size \\(M < N\\) at a slightly worse \\(O((N/M) \log M)\\) average computations per sample. |
|
|
|
|
|
|
|
|
|
|
|
### Impulse responses |
|
|
|
|
|
|
|
Sometimes we need to simulate non-rational transfer functions. |
|
|
|
Consider a general transfer function $H(f)$, written in terms of $f$ rather than $s$ for simplicity. |
|
|
|
Our original definition of a transfer function for an input $x(t)$ and output $y(t)$ is |
|
|
|
$$ Y(f) = H(f) X(f) $$ |
|
|
|
Consider a general transfer function \\(H(f)\\), written in terms of \\(f\\) rather than \\(s\\) for simplicity. |
|
|
|
Our original definition of a transfer function for an input \\(x(t)\\) and output \\(y(t)\\) is |
|
|
|
\\[ Y(f) = H(f) X(f) \\] |
|
|
|
and by applying the convolution theorem again, we obtain |
|
|
|
$$ y(t) = (h \ast x)(t) = \int_{-\infty}^\infty h(\tau) x(t - \tau) d\tau $$ |
|
|
|
where $h(t)$ is the *impulse response* of our filter. |
|
|
|
\\[ y(t) = (h \ast x)(t) = \int_{-\infty}^\infty h(\tau) x(t - \tau) d\tau \\] |
|
|
|
where \\(h(t)\\) is the *impulse response* of our filter. |
|
|
|
|
|
|
|
The signal $h(t)$ is the result of processing a [delta function](https://en.wikipedia.org/wiki/Dirac_delta_function) through our filter, since $\delta(t)$ is the "identity" function of convolution, i.e. $h(t) = (h \ast \delta)(t)$. |
|
|
|
Clapping your hands or popping a balloon (both good approximations of $\delta$) in a large cathedral will generate a very sophisticated impulse response, which can be recorded as $h(t)$ and processed in a FIR filter algorithm to reproduce arbitrary sounds as if they were performed in the cathedral. |
|
|
|
The signal \\(h(t)\\) is the result of processing a [delta function](https://en.wikipedia.org/wiki/Dirac_delta_function) through our filter, since \\(\delta(t)\\) is the "identity" function of convolution, i.e. \\(h(t) = (h \ast \delta)(t)\\). |
|
|
|
Clapping your hands or popping a balloon (both good approximations of \\(\delta\\)) in a large cathedral will generate a very sophisticated impulse response, which can be recorded as \\(h(t)\\) and processed in a FIR filter algorithm to reproduce arbitrary sounds as if they were performed in the cathedral. |
|
|
|
|
|
|
|
Repeating this process in the digital realm gives us the discrete convolution. |
|
|
|
$$ y_k = \sum_{n=-\infty}^\infty h_n x_{k-n} $$ |
|
|
|
\\[ y_k = \sum_{n=-\infty}^\infty h_n x_{k-n} \\] |
|
|
|
|
|
|
|
|
|
|
|
#### Brickwall filter |
|
|
|
|
|
|
|
An example of a non-rational transfer function is the ideal lowpass filter that fully attenuates all frequencies higher than $f_c$ and passes all frequencies below $f_c$. |
|
|
|
An example of a non-rational transfer function is the ideal lowpass filter that fully attenuates all frequencies higher than \\(f_c\\) and passes all frequencies below \\(f_c\\). |
|
|
|
Including [negative frequencies](https://en.wikipedia.org/wiki/Negative_frequency), its transfer function is |
|
|
|
$$ |
|
|
|
H(f) = \begin{cases} |
|
|
|
@@ -206,20 +206,20 @@ H(f) = \begin{cases} |
|
|
|
0 & \text{otherwise} |
|
|
|
\end{cases} |
|
|
|
$$ |
|
|
|
The inverse Fourier transform of $H(f)$ is |
|
|
|
$$ h(t) = 2 f_c \operatorname{sinc}(2 f_c t) $$ |
|
|
|
where $\operatorname{sinc}(x) = \sin(\pi x) / (\pi x)$ is the [normalized sinc function](https://en.wikipedia.org/wiki/Sinc_function). |
|
|
|
Although the impulse response is infinitely long, restricting it to a finite range $[-T, T]$ and shifting it forward by $T$ produces a finite causal impulse response that can be solved by a fast FIR algorithm to produce a close approximation of an ideal brickwall filter. |
|
|
|
The inverse Fourier transform of \\(H(f)\\) is |
|
|
|
\\[ h(t) = 2 f_c \operatorname{sinc}(2 f_c t) \\] |
|
|
|
where \\(\operatorname{sinc}(x) = \sin(\pi x) / (\pi x)\\) is the [normalized sinc function](https://en.wikipedia.org/wiki/Sinc_function). |
|
|
|
Although the impulse response is infinitely long, restricting it to a finite range \\([-T, T]\\) and shifting it forward by \\(T\\) produces a finite causal impulse response that can be solved by a fast FIR algorithm to produce a close approximation of an ideal brickwall filter. |
|
|
|
|
|
|
|
|
|
|
|
### Windows |
|
|
|
|
|
|
|
The impulse response $h_n$ is defined for all integers $n$, so it is both non-causal (requires future knowledge of $x(t)$ to compute $y(t)$) and infinitely long. |
|
|
|
The impulse response \\(h_n\\) is defined for all integers \\(n\\), so it is both non-causal (requires future knowledge of \\(x(t)\\) to compute \\(y(t)\\)) and infinitely long. |
|
|
|
|
|
|
|
*TODO* |
|
|
|
|
|
|
|
|
|
|
|
### To-do |
|
|
|
## To-do |
|
|
|
|
|
|
|
- digital filters |
|
|
|
- windows |
|
|
|
|
Andrew Belt
6 years ago
