Linear phase with a non-integer delay?

Question

Let's say you are given a frequency domain specification for a filter design.

As a simple example, let's use an ideal (brickwall) LPF. Now the impulse response of an ideal LPF is a $\operatorname{sinc}(\cdot)$ function which is infinite length and not causal. So to make it feasible you truncate it and shift it to the right. Now this shifting corresponds to a linear phase in the frequency response as follows:

$$H(e^{j\omega}) e^{-j\omega n_0}$$

Now if this $n_0$ is an integer then we do not have any problems in the time domain impulse response $h[n - n_0]$. But what if $n_0$ is not an integer which is entirely possible so how does that translate to the impulse response which cannot be defined for a non integer $n_0$ shift.

So what happens to the impulse response?

Answer 1

If you design a lowpass filter by windowing and shifting the impulse response of an ideal lowpass filter

$$h[n]=\frac{\sin(\omega_0n)}{\pi n}\tag{1}$$

then there's no other option than shifting the impulse response by an integer number of samples.

However, if for some reason you want a lowpass filter with a non-integer but constant delay, you could start with a delayed ideal lowpass filter:

$$h[n]=\frac{\sin(\omega_0(n-n_0))}{\pi (n-n_0)}\tag{2}$$

In this case the delay $n_0$ can be any real number. You can obtain an implementable filter by truncating/windowing the impulse response in $(2)$ such that it is causal and has a finite length. But note that in general the resulting filter won't have an exactly linear phase, but it will approximate a linear phase.

I designed six FIR filters with $100$ coefficients by simply truncating the delayed ideal impulse response in $(2)$ with $\omega_0=0.4\pi$ and with six different values for $n_0$: [49.0, 49.1, 49.2, 49.3, 49.4, 49.5]

The figure below shows the passband group delays of these six filters. Clearly, the filters with $n_0=49.0$ and $n_0=49.5$ samples have an exactly linear phase, while the others only approximate a linear phase (i.e., a constant group delay).

Note that in general the delay of a linear phase FIR filter can only be either an integer value of samples or an integer value plus $\frac12$. The value of that delay is determined by the filter length $N$ (the number of filter taps), and it equals $(N-1)/2$ samples.

Also take a look at this related question and its answers.

Answer 2

A fractional delay is just an integer delay at a sufficiently high sample rate.

Thus, the problem of fractional delay could be thought of as 1)Ideal upsampling (potentially to a continuous waveform), 2) delay, 3) Ideal downsampling (or sampling).

Thus ideal fractional delay could be a question of ideal resampling. Which (unfortunately) tends to involve sinc functions and infinite delay/complexity.

Answer 3

Recommended reading on the topic: https://ieeexplore.ieee.org/document/482137

Delaying is a linear operation and we express the process as convolution with a "delay filter", i.e.

$$h[n-x] = h[n]*\Delta[n-x]$$

where $\Delta[n-x]$ is the impulse response of a pure delay of "x". Turns out the impulse response of a pure delay is also a sinc function, i.e.

$$\Delta[n-x] = \text{sinc}[n-x]$$.

If the delay is an integer, (i.e. $ x = n_0, n_0 \in \mathbb{Z}$), than the this turns simply into a delayed unit impulse response (or Kronecker delta ) since the sinc for an integer argument is 0 except for $\text{sinc}[0] = 1$.

Fractional delays can be done, but they are awkward since (like an ideal lowpass filter) the impulse response is infinitely long and infinitely non-causal. Any practical application can only do an approximation using truncation and/or windowing using a complex trade off between latency, phase error, amplitude error, computational cost, etc.

Answer 4

I'd like to add a response as I just computed some stuff, which hopefully is correct, but everything adds up, so I am hopeful!

I assume a system response $H(e^{j\omega}) = e^{-j\phi(\omega)}$, so a generic allpass. Furthermore some generic input signal spectrum $X(e^{j\omega }) = \sum_{k=-\infty}^{\infty}x(k)e^{-j\omega k}$ and the corresponding output spectrum $ Y(e^{j\omega}) = X(e^{j\omega}) H(e^{j\omega})$.

We are interested in the output signal $y(n)$ and how it relates back to $x(n)$. First, I will not assume a special form of the phase response. Generally, we obtain

$y(n) = \frac{1}{2\pi} \int_{-\pi}^{\pi} Y(e^{j\omega})e^{j\omega n}d\omega = \frac{1}{2\pi} \int_{-\pi}^{\pi} e^{-j\phi(\omega)} \sum_{k=-\infty}^{\infty}x(k)e^{-j\omega k} e^{j\omega n}d\omega = \frac{1}{2\pi}\sum_{k=-\infty}^{\infty}x(k)\int_{-\pi}^{\pi} e^{-j\phi(\omega)} e^{j\omega (n-k)}d\omega$

Because $\phi(\omega)$ is periodic with period $2\pi$, we can expand $e^{-j\phi(\omega)}$ into a Fourier series, and actually, we see the computation of some fourier coeffients in the final integral.

For a function $f$ with a period of $2\pi$ and the complex fourier series, we have the general formular $ c_n = \frac{1}{2\pi}\int_{-\pi}^{\pi}f(\omega)e^{-j\omega n}d\omega$ for the fourier coefficients. I assume $f$ is such, that the integral exists.

Now we obtain further

$ \frac{1}{2\pi}\int_{-\pi}^{\pi} e^{-j\phi(\omega)} e^{j\omega (n-k)}d\omega = \overline{\frac{1}{2\pi}\int_{-\pi}^{\pi} e^{j\phi(\omega)} e^{-j\omega (n-k)}d\omega} = \overline{c_{n-k}(e^{j\phi(\omega)})}$

where $c_{n-k}(e^{j\phi(\omega)})$ denotes the fourier coefficients of $e^{j\phi (\omega)}$.

Now, using the relation $c_n(\overline{f}) = \overline{c_{-n}(f)}$, where $\overline{\cdot}$ denotes the complex conjugate, we obtain

$\overline{c_{n-k}(e^{j\phi(\omega)})} = \overline{c_{-(-n+k)}(e^{j\phi(\omega)})} = c_{k-n}(\overline{e^{j\phi(\omega)}}) = c_{k-n}(e^{-j\phi(\omega)})$

It follows

$ y(n) = \sum_{k=-\infty}^{\infty}x(k)c_{k-n}(e^{-j\phi(\omega)}).$

So we obtain $y(n)$ by convolving the input signal with the time reversed fourier coefficients of the system's transfer function. Now, this equation yields the well known result for a linear phase response (try it out, the fourier coefficients are obvious). However, if we have $\phi(\omega) = r\omega$ with real, noninteger $r$, we can compute, that this phase response indeed phase shifts the input signal by $r$ samples: the fourier coefficients of $e^{-jr \omega}$ are

$ c_n(e^{-j r\omega}) = (-1)^n \frac{sin(\pi r)}{\pi (n+r)}$.

Pluggin this into the previous equation, we get

$y(n) = \sum_{k=-\infty}^{\infty} x(k) c_{k-n}(e^{-j r\omega}) = \sum_{k=-\infty}^{\infty} x(k) (-1)^{k-n} \frac{sin(\pi r)}{\pi (k-n + r)}$

Using $ (-1)^{k-n}sin(\pi r) = sin(\pi r + \pi (k-n))$, we can actually see, that this formular is just shannon's interpolation formular with sampling period $T=1$ and some index changes. Therefore, if you filter an input signal with the system $H(e^{j\omega}) = e^{-j r \omega}$, it delays the input by an amount of $r$ samples, i.e., it is as if the input signal is interpolated using shannongs formular, shifted (delayed) in continous time domain by $r$, and then resampled with a period of $T=1$.

This is what some other people have said, and what you can find elsewhere on the internet, however, I have never seen it actually being calculated. It is missing from typical signal processing tables.

I hope I did not make a mistake :D

Answer 5

I don't think I had seen this question before today. If it's essentially how to find the coefficient set for a pure bandlimited delay of a non-integer number of samples, we first break the delay into the sum of an integer and fractional component. The integer component tells you which $L$ adjacent samples that are linearly combined and the fractional part tells you what the $L$ coefficients are to combine the samples. Even if the filter kernel is not a windowed-sinc function, that's the mechanism. Integer part - which samples. Fractional part - which coefficients.