Introduction to Transformation of Signals - Signals and Systems - Electrical

Table of Contents
1. Introduction
2. Fourier Series and Fourier Transform
3. Signals as Vectors and Countability
4. Dot Product (Inner Product) of Signals
5. Eigenvalue and Eigensignal of an LSI System
6. Conclusion
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Introduction

A fundamental idea in the analysis of signals and systems is the transformation of signals. A transform provides an alternative representation of a signal in a domain different from its natural domain. This alternate view often reveals properties that are not obvious in the original domain and simplifies analysis and design.

• The transformation of a signal from the time domain into a representation of its frequency components and phases is known as Fourier analysis. Fourier analysis decomposes signals into sinusoids (or complex exponentials) that are easier to study for many engineering problems.
• We use transformations because some tasks (for example filtering, stability analysis, spectral estimation, modulation) are easier to perform or understand in a transformed domain. Transformations are usually applied to an independent variable (time, space, etc.) to obtain a representation in a dual domain (frequency, wavenumber, etc.).

Examples:

1. A doctor examines a patient's heartbeat. The raw measurement is a time signal, but diagnostic displays (such as an ECG trace) convert the signal into spatial/graphical form to make patterns and abnormalities easier to detect.
2. For a musician, although sound occurs in time, the frequency content determines pitch and harmony; therefore frequency-domain representations are more useful for analysing musical notes and timbre.
3. For an electrical circuit, input and output voltages or currents are functions of time. An oscilloscope displays these time signals as spatial traces; frequency-domain tools (spectrum analysers) show the frequency content that is important for design and troubleshooting.

Fourier Series and Fourier Transform

• Every periodic signal can be expressed as a sum of sinusoidal functions whose frequencies are integer multiples of a base frequency called the fundamental frequency. This representation is the Fourier series.
• An aperiodic signal can be viewed as a periodic signal with period tending to infinity. As the period becomes infinite, the discrete set of harmonic frequencies becomes continuous and the summation in the Fourier series turns into an integral. This continuous-frequency representation is the Fourier transform.
• Treating signals as vectors in a function space is useful: a finite-dimensional vector is specified by its components along basis directions; similarly, a signal can be expanded in terms of basis functions (for Fourier series/transform the basis functions are sinusoids or complex exponentials).

Fourier Series (periodic signals)

For a periodic signal with period \(T_0\) and fundamental angular frequency \(\omega_0 = \dfrac{2\pi}{T_0}\), the complex exponential Fourier series representation is

where the coefficients \(C_k\) are given by

Fourier Transform (aperiodic signals)

The (continuous-time) Fourier transform of a signal \(x(t)\) is

If \(X(\omega)\) exists (in the usual sense), the inverse transform reconstructs \(x(t)\):

These transforms convert convolution in time to multiplication in frequency, and they make it straightforward to study filtering, modulation, and spectral properties.

Signals as Vectors and Countability

Signals can be treated as elements of vector spaces. A discrete-time signal corresponds to a sequence and can be thought of as a vector with countably infinite components; a continuous-time signal corresponds to a function defined over a continuum and can be thought of as a vector in an uncountably infinite-dimensional space (a function space).

Countable Infinity

• A set is countably infinite if its elements can be placed in one-to-one correspondence with the natural numbers (or with the integers). Examples: integers, rational numbers.
• The set of rational numbers is countable because each rational can be represented by a pair of integers (numerator and denominator), and such pairs can be enumerated.

Example: Prove that the set of real numbers is not countably infinite.

• Assume, for contradiction, that the set of real numbers in the interval \((0,1)\) is countable; then we can list them as \(r_1, r_2, r_3, \dots\).
• Construct a new real number \(r\) in \((0,1)\) whose decimal expansion differs from the decimal expansion of \(r_k\) at the \(k\)-th decimal digit (choose the \(k\)-th digit of \(r\) to be any digit different from the \(k\)-th digit of \(r_k\), avoiding 9 to prevent ambiguity).
• By construction, \(r\) differs from every listed \(r_k\) in at least one decimal place, so \(r\) is not in the list. This contradicts the assumption that all reals in \((0,1)\) were listed.
• Therefore, the real numbers are uncountable.

Note: A discrete signal \(x[n]\) is naturally associated with a countably infinite-dimensional vector space. A continuous-time signal \(x(t)\) belongs to an uncountably infinite-dimensional function space.

Dot Product (Inner Product) of Signals

The inner product (dot product) extends the geometric notion of projection and angle to signals. It is a binary operation that maps two signals into a scalar. Inner products let us define notions of length (energy), orthogonality, and projection for signals.

Finite-dimensional vectors

For two complex vectors \(X\) and \(Y\) in \(\mathbb{C}^N\), with components \(X = (x[1], x[2],\dots,x[N])\) and \(Y = (y[1], y[2],\dots,y[N])\), the inner product is

\[ \langle X, Y \rangle = \sum_{k=1}^{N} x[k]\, y^*[k] \]

Properties required for an inner product

• Conjugate symmetry: \(\langle x, y \rangle = \langle y, x \rangle^*\).
• Linearity: \(\langle a x_1 + b x_2, y \rangle = a \langle x_1, y \rangle + b \langle x_2, y \rangle\) for scalars \(a,b\).
• Positive-definiteness: \(\langle x, x \rangle \ge 0\) with equality if and only if \(x\) is the zero vector.

Inner product for signals

Continuous-time signals: For signals \(x(t)\) and \(y(t)\) for which the integral exists, the inner product is

\[ \langle x, y \rangle = \int_{-\infty}^{\infty} x(t)\, y^*(t)\, dt \]

Discrete-time signals: For sequences \(x[n]\) and \(y[n]\) for which the sum converges, the inner product is

\[ \langle x, y \rangle = \sum_{n=-\infty}^{\infty} x[n]\, y^*[n] \]

Eigenvalue and Eigensignal of an LSI System

The concepts of eigenvalue and eigensignal for systems are analogous to eigenvalues and eigenvectors of linear operators. In signals and systems, these arise naturally for linear, shift-invariant (LSI) systems.

Consider an LSI system characterised by its impulse response \(h(t)\). If an input \(x(t)\) produces an output \(y(t)\) that is the same as the input up to a scalar multiplier, i.e.

\[ y(t) = A\, x(t) \]

then \(x(t)\) is called an eigensignal of the system and \(A\) is the corresponding eigenvalue (a scalar, possibly complex).

For an LSI system the input-output relation is convolution:

Complex exponential as eigensignal

Consider the complex exponential input

\[ x(t) = e^{j\omega t} \]

The output of the LSI system is

Using the substitution \(\lambda = t-\tau\) (or by noting time invariance), the exponential factor can be taken outside the integral with respect to \(\lambda\), yielding

Define the system frequency response (eigenvalue) as

Therefore

Thus, the complex exponential \(e^{j\omega t}\) is an eigensignal of any LSI system (assuming the integral converges), and the corresponding eigenvalue is the frequency response value \(H(j\omega)\).

In inner-product language, the eigenvalue \(H(j\omega)\) can be regarded as the projection (inner product) of the impulse response \(h(t)\) onto the complex exponential \(e^{j\omega t}\) with complex-conjugation in the appropriate place:

This special property of complex exponentials-turning convolution into scalar multiplication-is a principal reason for representing signals in terms of complex exponentials (Fourier methods). Many system analyses and designs exploit this property to simplify calculations.

Conclusion

In this lecture you have learnt:

• Transforms provide alternative domains to examine signals. They are essential for understanding many signal properties; the Fourier transform is a central example.
• A discrete-time signal \(x[n]\) may be viewed as a vector in a countably infinite-dimensional space; a continuous-time signal \(x(t)\) belongs to an uncountably infinite-dimensional function space.
• We can define inner products for signals, introducing geometric concepts such as energy, length, orthogonality and projection. Inner products satisfy conjugate symmetry, linearity, and positive-definiteness.
• A complex exponential signal \(e^{j\omega t}\) is an eigensignal of a stable LSI system, and the corresponding eigenvalue (the system frequency response) equals the inner product between the impulse response and the complex exponential.