IEEE Microwave Magazine - May 2016 - 51

Shaping the Envelope to the Drain Voltage
After a proper representation of the envelope is computed, the signal will need to be mapped to a voltage
value for the PA. The trivial solution is a linear map from
the envelope representation into a voltage value. However, the voltage value could be further altered to force
certain envelope values to maintain a lower drain voltage
than the linear solution, thereby forcing the PA to operate
more often in the distortion region close to saturation to
increase the efficiency. Several factors affect the method
used for shaping the envelope to increase efficiency:
* the hardware limitation of the voltage modulator
itself (minimum voltage, maximum voltage, and
bandwidth/slew rate)
* the requirement that operation be above the transistor's knee voltage [6]
* the amount of efficiency required by the ET PA.
The ET PA's efficiency can be identified by sweeping
all possible combinations of drain voltage and output
powers available for the device. The designer could force
the ET PA to operate at the highest possible efficiency all
the time by shaping the envelope to give maximum efficiency for a given output power value. However, using
this shaping function introduces a wide bandwidth
requirement for the envelope amplifier [21], [24] that is
dependent on the bandwidth of the input signal. Some
shaping functions that attempt to maintain a minimum
drain voltage for a longer period of time (which results
in higher efficiency) include the following.

Linear Shaping Function [6]
A linear shaping function may be derived by using the
envelope as a linear slope between the minimum and
maximum voltage as follows:
M linear (n) = Vmin + (Vmax - Vmin) # E s (n) .

(15)

The shaping function may be optimized such that
the PA operates at a lower drain voltage up to a certain normalized input envelope value. An alternative

May 2016

0
Power Spectrum (dB)

dual-band envelope signal, and the slew-rate limited
envelope gives similar performance, except that the
slope of the voltage change is constrained by the slewrate parameter.
The frequency domain provides additional information on the combination functions, as shown in
Figure 5. It is apparent that the spectra of the average
envelope is significantly less at high frequencies compared to the other three.
Although most of the envelope spectra are located
in the range of dc to several kilohertz [22], the previously discussed functions can be further improved
by reducing the residual spectral components at the
input of the modulator through low-pass filtering [13],
smoothing functions [23], or the operation of two envelope-combination functions [20].

-20
-40
-60
-80
-100
-120

0

10

20
30
40
Frequency (MHz)

Es,peak

Es,avg

50

60

Es,sr

Dual-Band Envelope

Figure 5. The spectra of the dual-band envelope and some
of their low-frequency envelope representations.
linear formulation of the relationship between the
envelope and the drain voltage is [6]
Z Vmin
E s (n) 1 a
]]
(Vmax - Vmin)
[
M linear, a (n) =
(E s (n) - a) E s (n) $ a .
1 -a
]
+
V
min
\
(16)
A controllable parameter, a , denotes the maximum
normalized envelope magnitude where the drain voltage stays at a constant Vmin . A sharp transition point
exists for M linear, a , and some researchers have proposed
their own shaping functions to avoid this phenomenon.

Polynomial Function [25]
A convex polynomial shaping function can be used
to shape the envelope to the voltage and may be generalized as
M N (n) = (V Nmin + E s (n) N) 1/N .

(17)

A suitable compromise between efficiency and linearity for the shaping function occurs when the polynomial order N is set to 6 (N = 6) and is denoted as
M N6 (n) = (V 6min + E s (n) 6) 1/6 .

(18)

Wilson Function [26]
The basis of the Wilson shaping function is to utilize
the curvature of a cosine wave to shape the input envelope. The function can be mathematically described by
M wilson (n) = Vmin ` r j;1 - ` 2 j cos c E s (n) r - 2 mE .
2Vmin
r -2
r
(19)
Figure 6(a) shows the drain voltage for various envelope-shaping functions with respect to the normalized magnitude of the input envelope. Maintaining a

51



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