Noise Fundamentals

If you terminate a UUT input with a matched resistive load (typically 50 Ω) and measure the noise power density, in watts/Hz, at its output (N₀), the noise figure (NF) is given by the following equation:

where

• G is the power gain (in linear units) of the UUT
• k is Boltzmann's constant (≈ 1.38 ×10^-23 J/K)
• T is the ambient temperature (≈ 290° K)

If you use the vector signal analyzer to measure the output noise of a UUT, the resulting measurement contains both UUT noise and noise intrinsic to the vector signal analyzer. If the UUT gain (G_UUT in dB) is known, compute the noise figure of the UUT with the following equation:

$NF=10\log [\frac{N_m-N_{rfsa}}{kT}+1]-G_{UUT}$ $NF=10\log [\frac{N_m-N_{rfsa}}{kT}+1]-G_{UUT}$

where

• N _rfsa is the noise, in watts/Hz, measured by the vector signal analyzer when its input is terminated with a matched resistive load
• N _m is the measured noise, in watts/Hz, with the UUT attached
• G is the linear power gain
• NF is expressed in dB

The noise due to a matched resistive load (N_i) can be expressed as the following relation:

(N_i) = kTB watts

where

• k is Boltzmann's Constant (k = 1.38 x 10^-23J/K
• T is the resistor temperature in Kelvin
• B is the bandwidth in Hz

If B is set to 1 Hz, then N_i is equal to the output noise density in watts/Hz. For the system shown in the following figure, the output noise floor (N_O) is the combination of the input noise multiplied by the gain or loss of the system plus the internal noise of the system (N_n).

The internal noise of a system is therefore represented by the following equation:

N _n = N_iG – N_O

Harmonic distortion is inherent to devices and systems that possess nonlinear characteristics. The more nonlinear the device, the greater its harmonic distortion.

Harmonic distortion can be expressed as a power ratio or as a percentage. Use the following formula to express it as a power ratio:

P _HD = P_fund–P_harm(dBc)

where

• P _HD is the power of the harmonic distortion in dBc
• P _fund is the fundamental signal power in dB or dBm
• P _harm is the power of the harmonic of interest in dB or dBm

Convert power to voltage and use the following equation to express harmonic distortion as a percentage ratio:

You can calculate THD by summing the power in each of the harmonics and dividing by the total power of the fundamental. As a general principle, a signal becomes visibly distorted when the THD approaches -30 dB.

The equation for THD is:

As this equation suggests, the THD specification evaluates the power in harmonic spurs from the second through the n^th harmonic. In practice, typical signal generators feature THD specifications for harmonics two through six. The following figure shows an example plot of a generated signal and illustrates the concept of THD specification in signal generators.

The previous figure shows an example plot of a 20 kHz sinusoid generated with an arbitrary waveform generator. Notice the power levels of the harmonic spurs, indicated with circles. This generator features -77 dBc or better of THD for the second through sixth harmonics.

THD generally deteriorates as the generated signal increases in frequency. When working with a signal generator, consider the THD throughout the bandwidth of the generator.

If F₁ and F₂ are the frequencies of the two tones, the third-order distortion products occur on both sides of these tones at 2F₂ – F₁ and 2F₁ – F₂ . Assuming that the power levels of the two tones are equal, IMD₃ is the difference between the power of the fundamental signals and the third-order products, as defined in the following equation:

IMD ₃ = P_o – P_o3

where P_o3 is the power level of one of the output third-order products, and P_o is the power level of one of the fundamental tones. The math becomes more involved if the powers of the two tones are different.

After you calculate the IMD₃ using the preceding formula, you can calculate the unit under test (UUT) output third-order intercept point (OIP₃) using the following equation:

${OIP}_3=\frac{{IMD}_3}{2}+P_o=\frac{1}{2}(3P_o-P_{o3})\left(dB\right)$ ${OIP}_3=\frac{{IMD}_3}{2}+P_o=\frac{1}{2}(3P_o-P_{o3})\left(dB\right)$

The input third-order intercept point (IIP₃) is defined as:

IIP ₃ = (OIP₃ – G)

Where G is the gain of the device. The IIP₃ number quantifies the third-order linearity input referred of a device.

The following figure shows the relationship between the second- and third-order distortions and the linear output of a device.

The two tones injected into the UUT must be free from any third-order products. These two tones are combined, or summed, at or before the UUT input. If the two tones are not well isolated, they intermodulate with each other and cause distortion. A signal combiner with good input-to-input isolation is recommended to minimize distortion of the input tones.

An amplifier maintains a constant gain for low-level input signals. However, at higher input levels, the amplifier saturates and its gain decreases. The 1 dB gain compression point (P_{1 dB}) indicates the power level that causes the gain to drop by 1 dB from its small signal value.

The 1 dB gain compression point is derived from the gain relationship between output power and input power. Output power is plotted against input power in the following figure.

The straight line on this graph is an extrapolation of the small signal gain of the unit under test (UUT). The input 1 dB compression point is the input power that causes the UUT gain to drop by 1 dB from this small signal value. In this figure, the gain drop occurs at approximately -12 dBm.