Harmonics, IMD and IP3

The transfer function f(x) of an ideal linear amplifier is a straight line:

f(x) = k₀ + k₁x

Applying a sinusoidal input of amplitude A and frequency f:

x = Acos ωt
ω = 2πf

The output is an exact (scaled) replica of the input:

f(x) = k₀ + k₁Acos ωt

Real amplifiers are non-linear:

f(x) = k₀ + k₁x + k₂x² + k₃x³ + k₄x⁴ + k₅x⁵ + ...

The additional terms constitute distortion. The 2^nd and 3^rd are usually the most significant:

Distortion ≈ k₂x² + k₃x³

Harmonic Distortion

If x = Acos ωt the distortion is approximately

k₂A²cos² ωt + k₃A³cos³ ωt

Consider the trigonometric identity 2cos²θ = 1 + cos2θ

The square term is twice the input frequency plus DC.

Consider the identity 4cos³θ = 3cosθ + cos3θ

The cube term contains components at f and 3f.

Non-linearity produces harmonics at 2f, 3f, 4f, 5f... e.t.c.

Intermodulation Distortion

Now consider what happens when there are two input frequencies f₁ and f₂

ω₁ = 2πf₁
ω₂ = 2πf₂
x = A₁cos ω₁t + A₂cos ω₂t

The 2^nd order distortion products are:

k₂(A₁cos ω₁t + A₂cos ω₂t)² = k₂( A₁²cos² ω₁t + 2A₁A₂cos ω₁t cos ω₂t + A₂²cos² ω₂t)

The cos² terms are harmonic distortion i.e. 2f₁ and 2f₂ as demonstrated above. In addition, we also have the product:

2A₁A₂cos ω₁t cos ω₂t

This is called intermodulation distortion because the signals modulate one another (like in a mixer).

Consider the identity 2 cos(α) cos(β) = cos(α − β) + cos(α + β)

The 2^nd order products are the sum and difference frequencies f₁+f₂ and f₁-f₂.

It gets even more complicated when the 3^rd order products are included:

	Harmonic	Intermodulation
2^nd	2f₁ 2f₂	f₁±f₂
3^rd	3f₁ 3f₂	2f₁±f₂ 2f₂±f₁

There will be 4^th, 5^th and higher products, but 2f₁-f₂and 2f₂-f₁ are usually the most important because they're the strongest close to f₁ and f₂.

Third Order Intercept

In radio systems, we're most concerned with spurii within the IF passband such as the 3^rd order intermodulation products 2f₁-f₂and 2f₂-f₁.

Consider a system with two inputs of amplitude A

x = A cos ω₁t + A cos ω₂t

The wanted output is

f(x) = k₀ + k₁A cos ω₁t + k₁A cos ω₂t

The output amplitude is k₁A

By expanding k₃x³ the amplitude of the 2f₁-f₂and 2f₂-f₁ products can be shown to be ¾ k₃A³

For every 3dB increase in input, the wanted output rises by 3dB but the IMD3 products rise by 9dB because they're proportional to the cube of amplitude. The point where they would overtake is the third order intercept point (IP3). It's an imaginary point; real receivers overload first.

IP3 is an industry standard figure of merit used to rate the linearity of receivers and other signal processing systems. High IP3 equals good linearity. Good linearity equals immunity to intermodulation distortion caused by strong signals on adjacent frequencies.

IP3 is measured by applying sufficient input to reveal spurii without overloading. Two signal generators and a spectrum analyzer are required. The gap between the spurii and the wanted outputs should close by 6dB for a 3dB increase in input. The intercept point is calculated by extrapolating.

O = Wanted output (dBm)
S = 3^rd order spurious response (dBm)
I = Input level (dBm)
IP3 = 3^rd order input intercept (dBm)

Figures for good receivers exceed +30dBm. The spacing of the carriers used to perform the test should be quoted. This will typically be 10 or 20KHz. For a signal processing stage such as a mixer, which has gain or insertion loss, IP3 may be quoted relative to the input or the output. A mixer with a conversion gain of 10dB and output intercept of +18dBm has an input intercept of +8dBm.

Mathematical prediction of IMD products

Consider the identity:

2 cos(α) cos(β) = cos(α − β) + cos(α + β)

From this alone, we can derive:

2cos²θ = 1 + cos2θ

4cos³θ = 3cosθ + cos3θ

4cosθ cos²φ = 2cosθ + cos(θ − 2φ) + cos(θ + 2φ)

The transfer function of a non-linear system is:

f(x) = k₀ + k₁x + k₂x² + k₃x³ + k₄x⁴ + k₅x⁵ + ...

Apply tones of equal amplitude (A) to the input:

x = A cos ω₁t + A cos ω₂t

x² = A² (cos² ω₁t + 2 cos ω₁t cos ω₂t + cos² ω₂t)

x³ = A³ (cos³ ω₁t + 3 cos ω₁t cos² ω₂t + 3 cos² ω₁t cos ω₂t + cos³ ω₂t)

The expansion of x² is:

x² = A² (	cos² ω₁t +	= ½ . A² (	1 + cos 2ω₁t +
	2 cos ω₁t cos ω₂t +		2 cos (ω₁ − ω₂)t + 2 cos (ω₁ + ω₂)t +
	cos² ω₂t)		1 + cos 2ω₂t)

We have: DC, 2f₁, 2f₂, f₁±f₂.

The expansion of x³ is:

x³ = A³ (	cos³ ω₁t +	= ¼ . A³ (	3 cos ω₁t + cos 3ω₁t +
	3 cos ω₁t cos² ω₂t +		6 cos ω₁t + 3 cos (ω₁ − 2ω₂)t + 3 cos (ω₁ + 2ω₂)t +
	3 cos ω₂t cos² ω₁t +		6 cos ω₂t + 3 cos (ω₂ − 2ω₁)t + 3 cos (ω₂ + 2ω₁)t +
	cos³ ω₂t)		3 cos ω₂t + cos 3ω₂t)

We have: f₁, f₂, 3f₁, 3f₂, 2f₁±f₂, 2f₂±f₁. Note: x³ contains energy at the fundamental (input) frequencies.

The amplitudes of the IM3 products (2f₁ − f₂) and (2f₂ − f₁) are proportional to A³ :

³/₄ . k₃ . A³

The amplitudes of the fundamental tones f₁ and f₂ are:

k₁.A + ⁹/₄ . k₃ . A³

Intermodulation ratio, the ratio of tones to products, is typically 40-80dB when measuring IP3. At this relative power level:

k₁.A + ⁹/₄ . k₃ . A³ ≈ k₁.A

Raising input amplitude (A) by 1dB, increases the IM3 products by 3dB. The output tones are frequently stated to rise by 1dB, but this is an approximation - they may rise a fraction more: perhaps 1% more, depending on the intermod ratio. Note: this analysis ignores terms above 3^rd order.