Direct-digital synthesis (DDS) is a mature digital-signal-processing (DSP) technology that offers great flexibility and power for generating complex waveforms. One of the advanced waveforms within the realm of DDS creation (given a dual-accumulator architecture) is chirp or linear frequency-modulation (FM) signals. In contrast to larger and more expensive arbitrary waveform generators, DDS chirp sources can save power, size, and cost in critical designs.

The principles of DDS technology were formulated in the late 1960s. In almost the reverse of sampling theory, a DDS source produces digital samples of a sinewave by means of an accumulator and sine lookup table; these digital samples are converted to analog waveforms via a digital-to-analog converter (DAC) and filter (Fig. 1, left) . The number of digital bits in the process determines the ultimate resolution of the output waveform (Fig. 1, right) .

A variable phase ramp is achieved by means of an accumulator with W digital bits. The accumulator is a phase generator in which 2^W states represent 2π phase conditions. Of the accumulator's W bits, Q (most significant) of these bits ( usually Q << W) are connected to the sine lookup table, which generates D output bits (usually Q = D + 2 or D + 3, to ensure minimum quantization) of sine stored waveform, sampled data. This analog sampled signal is then filtered and smoothed by the output lowpass filter, according to the rules of sampling theory (see refs. 1-5).

A simple example may serve to demonstrate the properties of the sampled data as well as the concept of positive and negative frequencies. In this example, a bar is rotating at a rate of 1 Hz and illuminated by a flashlight (sampled) blinking at a rate of 10 Hz (Fig. 2). Each time the light flashes, the bar appears to rotate 36 deg. forward in a clockwise motion, although this interpretation is not conclusive. The bar can rotate at 1 Hz, or 11 Hz, or 21 Hz, or any frequency that is 10N + 1 Hz and be interpreted in a similar fashion under the strobe light. In addition, there is another set of infinite frequencies that are given by 10N - 1 Hz, rotating counter clockwise, that yield same results. Because they rotate counterclockwise, they are called negative frequencies, 180 deg. relative to the main set of frequencies. Generally, the set NF_s + F_f or NF_s- F_f (where Fs is the sampling frequency, F_f is the fundamental frequency, and N is the multiplication factor), generate the same sampled data response.

The set F_f and F_s - F_f is therefore a " couple" in sampled data (Fig. 3). Their amplitude decreases because the DAC generates a sample-and-hold (S/H) waveform and not a sampled "delta function," so the amplitude is scaled by the S/H transfer function given by sin(x)/x, where x is 1/Fs (and therefore goes to zero at integer multiples of the sampling frequency F_s = F_clock in Fig. 3).

Negative frequencies are real physical phenomena and not just a mathematical outcome of the Fourier transform. When a signal in the vicinity of F₀ is mixed with F₀ itself (Fig. 4, left), the noise bandwidth of the lowpass filter is twice its bandwidth (BW) because all sidebands around F₀ (±BW) will pass into the filter. Signals above F₀ will generate a positive output while frequencies below F₀ will generate a negative output (Fig. 4, right).

Since the electrical signal in a DDS is a vector, positive and negative phases are possible, and positive and negative frequencies. If F₀ + 1 or F₀- 1 are mixed with F₀, and the mixer output at 1 Hz displayed on an oscilloscope, it would be impossible to tell the difference between the two outputs. What is known is that the two signals are offset-by 180 deg. To identify them, two components of the vector must be displayed, hence the in-phase (I) and quadrature (Q) components.

The basic equations for a DDS, to generate sinewave outputs and change frequencies by changing the control input word based on a W-bit accumulator and a D-bit DAC (assuming Q > D + 1) include

the following:

F_out = W₁ × (F_clock/2^W)

where:

W₁ = the control input.

For example, a 48-b accumulator DDS, running at a clock frequency of 1000 MHz, has a frequency resolution of 10⁹/2⁴⁸ ~ 3.5 µHz.

Spurious signals, which traditionally have been a limitation of DDS technology, are approximately given by -6D (dBc), or about -70 dBc for the 12-b DAC. (This is true mainly for clock frequencies below F_clock/4; above this clock rate, DAC errors begin to dominate.) A DDS source's switching speed is given by approximately 3/BW, where B is the output bandwidth of the lowpass filter.

The output frequency of a DDS is practically limited to 40 to 45 percent of the clock frequency, since the source generates both F_out and F_clock - F_out and there are limitations on how to filter F_clock - F_out as F_out gets closer to F_clock/2. This is a natural outcome of the sampling theorem that states that sampling rate must be at least two samples per cycle.

The frequency command words W₁ and 2^W- W₁ will generate the same output frequency, although the two output signals will be offset by 180 deg. For an input command of W₁, the accumulator increments W₁, 2W₁, 3W₁...until it reaches 2^W which is a complete cycle and 2^W will then be subtracted from the sum (modulus 2π).

When controlling 2^W - W₁, the accumulator will almost always exceed its full state so the residue will be 2^W - W₁, 2(2^W - W₁), 3(2^W - W₁), etc...therefore: 2^W - W₁, 2^W - 2W₁, 2^W - 3W₁, etc. The absolute value of the phase increment (slope) is similar, but with opposite phase sign.