Signal Characteristics

Energy of a Signal

Theorem: Rayleigh’s Theorem
The total energy of a signal in the time domain is equal to the total energy of its Fourier transform in the frequency domain:

\[ \int_{-\infty}^{\infty} |x(t)|^2 \, dt = \int_{-\infty}^{\infty} |X(f)|^2 \, df \]

Definition: Signal Energy
The energy of a signal \(x(t)\) is defined as:

\[ \mathcal{E}_x = \int_{-\infty}^{\infty} |x(t)|^2 \, dt \]

Using Rayleigh’s Theorem, this can be expressed equivalently as:

\[ \mathcal{E}_x = \int_{-\infty}^{\infty} |X(f)|^2 \, df \]

Energy Decomposition for Bandpass Signals

Since \(X_+(f)\) and \(X_-(f)\) have no overlap (\(X_+(f)X_-(f) = 0\)), the total energy \(\mathcal{E}_x\) can be decomposed as:

\[ \mathcal{E}_x = \int_{-\infty}^{\infty} |X_+(f) + X_-(f)|^2 \, df \]

Expanding this:

\[ \mathcal{E}_x = \int_{-\infty}^{\infty} |X_+(f)|^2 \, df + \int_{-\infty}^{\infty} |X_-(f)|^2 \, df \]

Since the energy is equally distributed:

\[ \mathcal{E}_x = 2 \int_{-\infty}^{\infty} |X_+(f)|^2 \, df \]

\[ \mathcal{E}_x = 2\mathcal{E}_{x_+} \]

Relationship with Lowpass Equivalent Signal

For the lowpass equivalent signal \(x_l(t)\):

\[ \mathcal{E}_x = 2 \int_{-\infty}^{\infty} \left|\frac{X_l(f)}{2}\right|^2 \, df \]

Simplifying:

\[ \mathcal{E}_x = \frac{1}{2} \mathcal{E}_{x_l} \]

Key Insight

The energy in the lowpass equivalent signal (\(x_l(t)\)) is twice the energy in the bandpass signal (\(x(t)\)). This relationship highlights how the energy is distributed between the real and imaginary components of the complex lowpass representation.

Inner Product

Definition: Inner Product of Two Signals
The inner product of two signals \(x(t)\) and \(y(t)\) is defined as:

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

Energy Re-expressed
The energy of a signal \(x(t)\) can be expressed using the inner product:

\[ \mathcal{E}_x = \langle x(t), x(t) \rangle \]

For two bandpass signals \(x(t)\) and \(y(t)\) with lowpass equivalents \(x_l(t)\) and \(y_l(t)\) (with respect to the same \(f_0\)):

\[ \langle x(t), y(t) \rangle = \frac{1}{2} \text{Re}\big[\langle x_l(t), y_l(t) \rangle\big] \]

Cross-Correlation Coefficient

Definition: Cross-Correlation Coefficient
The cross-correlation coefficient \( \rho_{x,y} \) between two signals \(x(t)\) and \(y(t)\) is given by:

\[ \rho_{x,y} = \frac{\langle x(t), y(t) \rangle}{\sqrt{\mathcal{E}_x \mathcal{E}_y}} \]

This represents the normalized inner product between the two signals.

From the relation \(\mathcal{E}_x = 2\mathcal{E}_{x_l}\), we can deduce that for bandpass signals \(x(t)\) and \(y(t)\) with the same \(f_0\):

\[ \rho_{x,y} = \text{Re}(\rho_{x_l, y_l}) \]

Proof: \( \rho_{x,y} = \text{Re}(\rho_{x_l, y_l}) \)

Expressing Bandpass Signals in Terms of Lowpass Equivalents

Given:

\[ x(t) = \Re \big[x_l(t)e^{j2\pi f_0 t}\big], \quad y(t) = \Re \big[y_l(t)e^{j2\pi f_0 t}\big] \]

where \( x_l(t) \) and \( y_l(t) \) are the complex lowpass (baseband) equivalents of the bandpass signals centered around carrier frequency \( f_0 \).

Defining the Cross-Correlation Coefficient

\[ \rho_{x,y} = \frac{\langle x(t), y(t) \rangle}{\sqrt{\mathcal{E}_x \mathcal{E}_y}}, \]

where

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

Substituting \( x(t) \) and \( y(t) \) into the Inner Product

\[ \langle x(t), y(t) \rangle = \int_{-\infty}^\infty \Re \big[x_l(t)e^{j2\pi f_0 t}\big] \cdot \Re \big[y_l(t)e^{j2\pi f_0 t}\big] \, dt \]

Expanding the Real Parts

Using the identity for the product of real parts:

\[ \Re[A] \cdot \Re[B] = \frac{1}{2} \Re[A B^* + A^* B] \]

Assuming \( x(t) \) and \( y(t) \) are analytic signals (containing only positive frequency components), i.e., \( x(t) \to x_+(t), y(t) \to y_+(t)\) after Hilbert transformation, the cross term \( \Re[A^* B] \) averages to zero over infinite time. Therefore:

\[ \langle x(t), y(t) \rangle = \frac{1}{2} \int_{-\infty}^\infty \Re \big[x_l(t)e^{j2\pi f_0 t} \cdot (y_l(t)e^{j2\pi f_0 t})^*\big] \, dt \]

Simplifying the Conjugate Term

\[ (y_l(t)e^{j2\pi f_0 t})^* = y_l^*(t)e^{-j2\pi f_0 t} \]

Substituting back:

\[ \langle x(t), y(t) \rangle = \frac{1}{2} \int_{-\infty}^\infty \Re \big[x_l(t)y_l^*(t)e^{j2\pi f_0 t}e^{-j2\pi f_0 t}\big] \, dt = \frac{1}{2} \int_{-\infty}^\infty \Re \big[x_l(t)y_l^*(t)\big] \, dt \]

Relating to the Lowpass Inner Product

The lowpass inner product is:

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

Thus:

\[ \langle x(t), y(t) \rangle = \frac{1}{2} \Re \big[\langle x_l(t), y_l(t) \rangle\big] \]

Substituting into the Cross-Correlation Coefficient

\[ \rho_{x,y} = \frac{\frac{1}{2} \Re \big[\langle x_l(t), y_l(t) \rangle\big]}{\sqrt{\mathcal{E}_x \mathcal{E}_y}} \]

Relating Bandpass Energy to Lowpass Energy

The energy of a bandpass signal \( \mathcal{E}_x \) is half the energy of its lowpass equivalent \( \mathcal{E}_{x_l} \), i.e.,

\[ \mathcal{E}_x = \frac{1}{2} \mathcal{E}_{x_l}, \quad \mathcal{E}_y = \frac{1}{2} \mathcal{E}_{y_l} \]

Thus:

\[ \sqrt{\mathcal{E}_x \mathcal{E}_y} = \sqrt{\left(\frac{1}{2}\mathcal{E}_{x_l}\right)\left(\frac{1}{2}\mathcal{E}_{y_l}\right)} = \frac{1}{2}\sqrt{\mathcal{E}_{x_l} \mathcal{E}_{y_l}} \]

Simplification using Energy Relationship

Substituting the corrected energy relationship into \( \rho_{x,y} \):

\[ \rho_{x,y} = \frac{\frac{1}{2} \Re \big[\langle x_l(t), y_l(t) \rangle\big]}{\frac{1}{2}\sqrt{\mathcal{E}_{x_l} \mathcal{E}_{y_l}}} = \Re \bigg[\frac{\langle x_l(t), y_l(t) \rangle}{\sqrt{\mathcal{E}_{x_l} \mathcal{E}_{y_l}}}\bigg] = \Re (\rho_{x_l, y_l}) \]

Thus, we conclude that the cross-correlation coefficient of the bandpass signals \( x(t) \) and \( y(t) \) is equal to the real part of the cross-correlation coefficient of their lowpass equivalents \( x_l(t) \) and \( y_l(t) \):

\[ \rho_{x,y} = \text{Re}(\rho_{x_l, y_l}) \]

\(\blacksquare\)

Orthogonal Signals

Definition: Orthogonal Signals
Two signals are said to be orthogonal if their inner product is zero:

\[ \langle x(t), y(t) \rangle = 0 \quad \implies \quad \rho_{x,y} = 0 \]

Key Insight

• Orthogonality in the baseband implies orthogonality in the passband.

• However, the converse is not necessarily true. Orthogonality in the passband does not guarantee orthogonality in the baseband.

Summary Table

We summary in the following table of all the signal representations and their associated notations, meanings, and mathematical descriptions.

Notation	Meaning	Mathematical Description
\(x(t)\)	Original bandpass signal	The real-valued time-domain representation of the signal.
\(\hat{x}(t)\)	Hilbert transform of \(x(t)\)	\(\hat{x}(t) = \frac{1}{\pi t} * x(t)\) (introduces \(-\pi/2\) phase shift to positive frequencies and \(+\pi/2\) to negative).
\(x_+(t)\)	Analytic signal (pre-envelope)	\(x_+(t) = \frac{1}{2}x(t) + j\frac{1}{2}\hat{x}(t)\)
\(x_l(t)\)	Lowpass equivalent (complex envelope)	\(x_l(t) = (x(t) + j\hat{x}(t))e^{-j2\pi f_0t}\)
\(x_i(t)\)	In-phase component	\(x_i(t) = x(t)\cos(2\pi f_0t) + \hat{x}(t)\sin(2\pi f_0t)\)
\(x_q(t)\)	Quadrature component	\(x_q(t) = \hat{x}(t)\cos(2\pi f_0t) - x(t)\sin(2\pi f_0t)\)
\(r_x(t)\)	Envelope (magnitude in polar coordinates)	\(r_x(t) = \sqrt{x_i^2(t) + x_q^2(t)}\)
\(\theta_x(t)\)	Phase (angle in polar coordinates)	\(\theta_x(t) = \arctan\left(\frac{x_q(t)}{x_i(t)}\right)\)
\(X(f)\)	Spectrum of the bandpass signal	Fourier transform of \(x(t)\), with frequency support around \(\pm f_0\).
\(X_+(f)\)	Positive-frequency spectrum	\(X_+(f) = X(f)u_{-1}(f)\), where \(u_{-1}(f)\) is the unit step function for \(f > 0\).
\(X_-(f)\)	Negative-frequency spectrum	\(X_-(f) = X(f)u_{-1}(-f)\).
\(X_l(f)\)	Spectrum of the lowpass equivalent signal	\(X_l(f) = 2X(f + f_0)u_{-1}(f + f_0)\)

Key Relationships

Bandpass Signal to Analytic Signal:

\[ x_+(t) = \frac{1}{2}x(t) + j\frac{1}{2}\hat{x}(t) \]

Analytic Signal to Lowpass Equivalent:

\[ x_l(t) = 2x_+(t)e^{-j2\pi f_0t} = (x(t) + j\hat{x}(t))e^{-j2\pi f_0t} \]

Polar Coordinates:

\[ x_l(t) = r_x(t)e^{j\theta_x(t)}, \quad x(t) = r_x(t)\cos(2\pi f_0t + \theta_x(t) \]

In-phase and Quadrature Components:

\[ x_l(t) = x_i(t) + jx_q(t) \]

Reconstruction of \(x(t)\) and \(\hat{x}(t)\)

Using the in-phase (\(x_i(t)\)) and quadrature (\(x_q(t)\)) components, the original signal \(x(t)\) and its Hilbert transform \(\hat{x}(t)\) can be expressed as:

\[ x(t) = x_i(t)\cos(2\pi f_0t) - x_q(t)\sin(2\pi f_0t) \]

\[ \hat{x}(t) = x_q(t)\cos(2\pi f_0t) + x_i(t)\sin(2\pi f_0t) \]