AD9786BSVRL datasheet(32/56 Pages) AD

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AD9786

Rev. B | Page 32 of 56

REAL AND COMPLEX SIGNALS

A complex signal contains both magnitude and phase

information. Given two signals at the same frequency, if the

leading signal in phase is cosinusoidal and the lagging signal is

sinusoidal, information pertaining to the magnitude and phase of

a combination of the two signals can be derived; the combination

of the two signals can be considered a complex signal. The cosine

and sine can be represented as a series of exponentials, recalling

that a multiplication by j is a counterclockwise rotation about

the Re/Im plane. The phasor representation of a complex signal

with Frequency f is shown in Figure 58.

A/2

FREQUENCY

–f

πft

C = Ae2πft = Acos(2

πft) + jAsin(2πft)

Acos(2

πft) = A

[e+j2πft + e–j2πft]

e+j2πft + e–j2πft

Asin(2

πft) = A

[je+j2πft + e–j2πft]

e+j2πft + e–j2πft

Figure 58. Complex Phasor Representation

The cosine term—referred to as the real in-phase, or I component,

of a complex signal—represents a signal on the real plane with

mirror symmetry about dc. The sine term—referred to as the

imaginary quadrature, or Q complex signal component—

represents a signal on the imaginary plane with mirror

asymmetry about dc.

The AD9786 has two channels of interpolation filters, allowing

both I and Q components to be shaped by the same filter transfer

function. The interpolation filter’s frequency response is a real

transfer function. Two DACs are required to represent a complex

signal. A single DAC can only synthesize a real signal. When a

DAC synthesizes a real signal, negative frequency components

fold onto the positive frequency axis. If the input to the DAC is

mirrored symmetrically about dc, the negative frequency

components fold directly onto the positive frequency compo-

nents in phase-producing, constructive signal summation. If

the input to the DAC is not mirrored symmetrically about dc,

negative frequency components might not be in phase with

positive frequency components, causing destructive signal

summation. Different applications might benefit from either

type of signal summation.

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