Signals and Domains

We use in this document a convenient convention that is more popular among physicists than it is among mathematicians. In a given application domain – for us, that is digital signal processing – when an object has several equivalent representations as functions, we use the same name for the object, and distinguish the representations by (a superset of the) domain of definition of the function. To every such domain we also associate fixed variable names.

In this document, we are dealing with discrete-time signals with a sample period \(\Delta t\) (or sample rate \(\Delta f = 1 / \Delta t\)). A signal \(x\) is is represented in the time domain as a function \(x(t)\) where \(t \in \mathbb{Z}\Delta t,\) in the frequency (or Fourier) domain as the function \(x(f)\) where \(f \in \mathbb{R}/{\Delta f},\) and in the complex (or Laplace) domain as a function \(x(z)\) where \(z \in \mathbb{C}.\) We often implicitly favor some representation and refer for example to \(x(t)\) as “the signal” instead of “the representation of the signal \(x\) in the time domain”.

If \(t\) is a free variable, \(x(t)\) denotes a function of the time \(t,\) if it is bound to some value, the value of the function. If there is some ambiguity in the choice of the representation, we use an assignment syntax, for example \(x(t=0)\) instead of \(x(0),\) because it could be mistaken as \(x(f=0).\)

Sets and Functions

The set of functions from \(A\) to \(B\) is denoted \(A \to B.\)

The set \(\mathbb{C}\) is the complex plane; \(\mathbb{U}\) refers to the unit circle centered on the origin: \[ \mathbb{U} = \{z \in \mathbb{C} \, , \, |z|=1 \}. \] The symbol \(\partial \mathbb{U}\) denotes the boundary of \(\mathbb{U}.\) Its positively oriented boundary, the closed path \(t \in [0,1] \mapsto e^{i2\pi t}\), is denoted \([\circlearrowleft]\).

For any \(r>0,\) \(D_r\) is the open disk with radius \(r\) centered on the origin: \[ D_r = \{z \in \mathbb{C} \, , \, |z| < r\} \] and for any \(r \in \left[0,1\right[,\) \(A_r\) is the open annulus with internal radius \(r\) and external radius \(1/r\): \[ A_r= \{z \in \mathbb{C} \, , \, r < |z| < 1/r \}. \]

Iverson Bracket

We¹ denote \(\{\,\cdot \,\}\) the function defined by: \[ \{b\} = \left| \begin{array}{ll} 1 & \mbox{if $b$ is true,} \\ 0 & \mbox{if $b$ is false.} \end{array} \right. \]

This elementary notation supercedes many other ones. For example, we can use \(\{x \in A\}\) instead of \(\chi_A(x)\) to denote the characteristic function of the set \(A,\) \(\{i=j\}\) instead of \(\delta_{ij}\) to denote the Kronecker delta, \(\{t \geq 0\}\) instead of \(H(t)\) to denote the Heaviside function.

In the sequel, all signals are discrete-time, hence we often drop this qualifier. Also, in this introductory section, although many definitions and results are valid in a more general setting, for the sake of simplicity, we always assume that signals are finite:

Fourier Transform

\(z\)-Transform

There is yet another useful representation of a finite signal – this time as a function of a complex variable \(z\) – and it is closely related to the frequency-domain representation.

Convolution and Filters

Given a quickly decreasing signal \(x(t)\) in the time domain, as in the finite signal case, its representation in the frequency domain is \[ x(f) = \Delta t \sum_{t \in \mathbb{Z}\Delta t} x(t) e^{i 2 \pi f t} \] and in the complex domain \[ x(z) = \Delta t \sum_{n \in \mathbb{Z}} x(t = n \Delta t) z^{-n}. \] However, the sums are not finite anymore; we consider that the values of the functions \(x(f)\) and \(x(z)\) are well defined when the sums are absolutely summable.

1. a quickly decreasing function \(x(t),\)

2. a holomorphic function \(x(z)\) defined on some neighbourhood of \(\mathbb{U},\)

3. a \(\Delta f\)-periodic and analytic function \(x(f)\) on \(\mathbb{R}.\)

If \(x(z)\) is holomorphic in a neighbourhood of the unit circle, then \(x(f) = x(z=e^{i 2 \pi f \Delta t})\) is \(\Delta f\)-periodic and analytic. Conversely, if \(x(f)\) analytic and \(\Delta f\)-periodic, it has a holomorphic extension, that we still denote \(x(f),\) in some open neighbourhood \(V\) of \(\mathbb{R}\) in \(\mathbb{C}.\) We can always ensure that \(x(f)\) is actually defined on as a tubular neighbourhood \(V_{\epsilon}\) of \(\mathbb{R}\) for some \(\epsilon >0,\) where \[ V_{\epsilon} = \mathbb{R} + D_{\epsilon} = \{ f\in \mathbb{C} \, , \, |\mathrm{Im} \, f| < \epsilon\}. \] Indeed, we can select \(\epsilon > 0\) such that \(U_{\epsilon} = [-\Delta f/2, +\Delta f/2] + D_{\epsilon}\) is included in \(V,\) and define a new analytic extension \(x'(f)\) on \(V_{\epsilon}\) by \(x'(f) = x(f - k\Delta f)\) where \(k \in \mathbb{Z}\) is such that \(f - k\Delta f \in U_{\epsilon}\) (by the isolated zeros theorem, this definition is unambigous). Let \(\sigma = 2 \pi \epsilon;\) for any \(z \in A_{\rho}\) with \(\rho = e^{-\sigma \Delta t},\) \(\phi(z) = x(f = (\log z) / i2\pi\Delta t)\) is independent of the determination of \(\log z\) ; it is holomorphic and \(x(f) = \phi(z=e^{i2\pi f \Delta t}).\)

Finally, starting from \[ x(t = n \Delta t) = \frac{1}{i 2 \pi} \int_{[\circlearrowleft]} \frac{x(z)}{\Delta t} z^{n-1} \, dz, \] we may introduce the path \(f \mapsto e^{i2 \pi f \Delta t}\) for \(f \in [-\Delta f/2, \Delta f /2],\) and we get \[ x(t = n \Delta t) = \frac{1}{i 2 \pi} \int_{-\Delta f/2}^{+\Delta f/2} \frac{x(z=e^{i 2 \pi f \Delta t})}{\Delta t} e^{i 2\pi f(n-1)\Delta t} (i 2 \pi \Delta t e^{i 2 \pi f\Delta t}) df, \] which after simplifications, yields for any \(t \in \mathbb{Z}\Delta t,\) \[ x(t) = \int_{-\Delta f/2}^{+\Delta f/2} x(f) e^{i 2\pi f t}df, \] the expected result. ‌\(\blacksquare\)

from numpy import *

df = 8000.0
dt = 1.0 / df
N = 1000
f = linspace(0.0, 0.5 * df, N)
z_f = exp(1j * 2 * pi * f * dt)
h_f = z_f / (z_f - 0.5)

Fourier Transform

Standard Defining Function

The example of Fourier transform \(x(f)\) computed in the previous section exhibited a very specific structure that is actually shared by all periodic hyperfunctions:

Non-Standard Defining Functions

If the complex number \(z\) satisfies \(|z|>1,\) we have \[ x_-(z) = - \Delta t \sum_{n > 0} \left[ \int_{-\Delta f / 2}^{+\Delta f / 2} x(f) e^{i 2 \pi f n \Delta t} \, df \right] z^{-n}, \] and the series \[-\Delta t \sum_{n > 0} x(f) e^{i 2 \pi f n \Delta t} z^{-n} = -x(f) \Delta t \sum_{n >0} (z^{-1} e^{i 2 \pi f m \Delta t})^n\] converges as a locally integrable function of \(f\) to \[-x(f) \Delta t (z^{-1} e^{i 2 \pi f \Delta t}) / (1 - z^{-1} e^{i 2 \pi f \Delta t}) = x(f) \Delta t / (1 - z e^{-i 2 \pi f \Delta t}),\] hence the formula for \(x_{\pm}(z)\) of the theorem also holds for \(|z| > 1.\) ‌\(\blacksquare\)

The time-domain representation of this filter is easy to determine: we have

\[ x(t) = \int_{-\Delta f / 2}^{+\Delta f / 2} x(f) e^{i 2\pi f t} \, df = \int_{-\Delta f / 4}^{+\Delta f / 4} e^{i 2\pi f t} \, df \] hence \[ x(t = n \Delta t) = \left[ \frac{e^{i 2\pi f n \Delta t}}{i 2\pi n \Delta t} \right]_{-\Delta f / 4}^{+ \Delta f / 4} = \mathrm{sinc} \, \frac{\pi n}{2}. \]

Linear Combination

As the Fourier transform and the \(z\)-transforms are linear operators, the multiplication of signals by a complex scalar and sum of signals can be defined in the time domain, by \[ (\lambda x)(t) = \lambda x(t), \; (x + y)(t) = x(t) + y(t), \] or equivalently in the frequency domain \[ (\lambda x)_r(f) = \lambda x_r(f), \; (x + y)_r(f) = x_r(f) + y_r(f), \] as well as in the complex domain \[ (\lambda x)_{\pm}(z) = \lambda x_{\pm}(z), \; (x + y)_{\pm}(z) = x_{\pm}(z) + y_{\pm}(z). \]

Modulation

Let \(x(t)\) be a signal, \(f_0 \in \mathbb{R}\) and \[ y(t) = x(t) e^{i2\pi f_0 t}. \] Straighforward computations show that \[ y_r(f) = x_r(f - f_0) \] and \[ y_{\pm}(z) = x_{\pm}(z e^{-i2\pi f_0 \Delta t}) \]

Integration (Frequency Domain)

Let \(x(f)\) be a \(\Delta f\)-periodic hyperfunction. It would be natural to define the integral of \(x(f)\) over one period as the limit when \(r\uparrow 1\) of \[ \int_{-\Delta f/2}^{+\Delta f/2} x(f) \, df = \lim_{r \uparrow 1} \int_{-\Delta f/2}^{+\Delta f/2} x_r(f) \, df \] but does this definition make sense? Are we sure that the limit always exists to begin with? Actually, it does and in a quite spectacular way: the integral under limit is eventually independent of \(r\):

Differentiation (Frequency Domain)

Let \(x(f)\) be a \(\Delta f\)-periodic hyperfunction. For every \(r \in \left[0,1\right[,\) the function \(x_r(f)\) is differentiable with respect to \(f.\) It would be natural to define the derivative of \(x(f)\) with respect to \(f\) by \[ \frac{dx(f)}{df} = \frac{\partial x_r(f)}{\partial f}. \] and then every periodic hyperfunction would be differentiable. But does this definition make sense? Is \(dx(f)/df\) well-defined as a hyperfunction?

Convolution (Time Domain), Product (Frequency Domain)

Fourier Inversion Formula

We already know enough about operational calculus of hyperfunctions to prove some interesting results. For example, we may now deal with the extension of the Fourier Inversion Formula to slowly increasing signals (in the time domain) or hyperfunctions (in the frequency domain).