Amplitude Scaling

Introduction

Amplitude scaling is the most fundamental signal operation — multiplying every value of a signal by a constant. When you turn up the volume on a speaker, you are amplitude-scaling the audio signal. When a voltage divider reduces a sensor output before feeding it to an ADC, that is amplitude scaling with an attenuation factor. Despite its simplicity, this operation has important consequences for energy calculations, system gain analysis, and signal normalization that every engineer must understand thoroughly.

Amplitude scaling preserves the shape of the signal entirely. The waveform looks identical except for a change in its vertical extent. No frequencies are created or destroyed, no time shifts are introduced, and the fundamental character of the signal remains unchanged. This makes it a linear operation in the truest sense.

Mathematical Definition

For a continuous-time signal: $$y(t) = c \cdot x(t)$$

For a discrete-time signal: $$y[n] = c \cdot x[n]$$

where $c$ is a real constant called the scaling factor, gain, or attenuation depending on its value:

• $|c| > 1$: Amplification — the signal magnitude increases
• $|c| < 1$: Attenuation — the signal magnitude decreases
• $c < 0$: Inversion — the signal polarity flips (values above zero go below, and vice versa)
• $c = -1$: Pure inversion — same magnitude, opposite sign at every point
• $c = 0$: Annihilation — the signal is destroyed entirely (trivial case)

Effect on Signal Energy

The energy of a continuous-time signal is defined as $E_x = \int_{-\infty}^{\infty}|x(t)|^2 dt$. After amplitude scaling:

$$E_y = \int_{-\infty}^{\infty}|cx(t)|^2 dt = c^2 \int_{-\infty}^{\infty}|x(t)|^2 dt = c^2 E_x$$

This is a crucial result: energy scales as the square of the amplitude factor. Doubling the amplitude quadruples the energy. Halving the amplitude reduces energy to one-quarter.

For discrete-time signals, the same relationship holds: $$E_y = \sum_{n=-\infty}^{\infty}|cx[n]|^2 = c^2 \sum_{n=-\infty}^{\infty}|x[n]|^2 = c^2 E_x$$

Effect on Signal Power

For power signals (infinite duration, finite average power):

$$P_y = \lim_{T\to\infty}\frac{1}{2T}\int_{-T}^{T}|cx(t)|^2 dt = c^2 P_x$$

The same quadratic relationship applies. This has practical implications: if you amplify a radio signal by a factor of 10 ($c = 10$), the transmitted power increases by a factor of 100.

Effect on the Frequency Spectrum

Applying the Fourier Transform: $$\mathcal{F}\{cx(t)\} = c \cdot X(j\omega)$$

The spectrum shape is completely unchanged — only its magnitude is scaled vertically by the factor $c$. No new frequency components are introduced. The phase spectrum also remains identical. This is a direct consequence of the linearity of the Fourier Transform.

Similarly for the Laplace and Z-Transforms: $$\mathcal{L}\{cx(t)\} = c \cdot X(s) \qquad \mathcal{Z}\{cx[n]\} = c \cdot X(z)$$

Effect on RMS Value

The root mean square value scales linearly: $$x_{rms,y} = |c| \cdot x_{rms,x}$$

This is because the RMS value is the square root of the mean squared value, and the $c^2$ inside the square root becomes $|c|$ outside.

The Decibel Scale

Because power scales quadratically with amplitude, engineers use the logarithmic decibel scale to express gain:

Power gain in decibels: $$G_{dB} = 10\log_{10}\left(\frac{P_y}{P_x}\right) = 10\log_{10}(c^2) = 20\log_{10}|c|$$

Voltage/amplitude gain in decibels: $$G_{dB} = 20\log_{10}\left(\frac{|y|}{|x|}\right) = 20\log_{10}|c|$$

The decibel scale converts multiplicative gains into additive ones, making cascaded system analysis straightforward.

Common Reference Values

The -3 dB point is particularly important in filter design, as it marks the cutoff frequency where the output power drops to half the passband power.

Worked Example

Problem: A signal $x(t) = 5\sin(100\pi t)$ is passed through an amplifier with gain $c = 4$ and then through an attenuator with factor $c = 0.1$. Find the overall output, its energy density spectrum, and the overall gain in dB.

Solution:

• After amplifier: $y_1(t) = 4 \times 5\sin(100\pi t) = 20\sin(100\pi t)$
• After attenuator: $y(t) = 0.1 \times 20\sin(100\pi t) = 2\sin(100\pi t)$
• Overall scaling: $c_{total} = 4 \times 0.1 = 0.4$
• Overall gain: $20\log_{10}(0.4) = 20 \times (-0.3979) = -7.96$ dB
• Energy ratio: $c_{total}^2 = 0.16$ (output energy is 16% of input energy)

Practical Applications

Automatic Gain Control (AGC): In communication receivers, AGC circuits dynamically adjust the gain $c$ to maintain a constant output level despite varying input signal strength. The scaling factor is continuously adapted based on the measured output power.

Normalization: Before processing, signals are often normalized to have unit peak value or unit energy: $$x_{normalized}(t) = \frac{x(t)}{\max|x(t)|} \quad \text{(peak normalization)}$$ $$x_{normalized}(t) = \frac{x(t)}{\sqrt{E_x}} \quad \text{(energy normalization)}$$

Voltage Dividers: A resistive voltage divider implements amplitude scaling with: $$c = \frac{R_2}{R_1 + R_2}$$

This always gives $0 < c < 1$ (attenuation only).

Digital Audio: In audio processing, amplitude scaling by values greater than 1.0 can cause clipping if the scaled values exceed the representable range, introducing distortion.

Key Takeaways

• Amplitude scaling $y(t) = cx(t)$ preserves signal shape while changing magnitude
• Energy and power scale as $c^2$ — the quadratic relationship is fundamental
• The frequency spectrum shape is unchanged; only magnitude is scaled by $c$
• Decibels convert multiplicative gain to additive: $G_{dB} = 20\log_{10}|c|$
• Negative $c$ inverts polarity; $|c| > 1$ amplifies; $|c| < 1$ attenuates
• Cascaded amplitude scaling factors multiply: $c_{total} = c_1 \cdot c_2 \cdot c_3 \cdots$
• The -3 dB point corresponds to $c = 1/\sqrt{2} \approx 0.707$ (half-power)