# About Spectrum Analysis

Spectrum analyzers and Fourier transformers may be assembled from sets of Dispersive Delay Lines (DDL). The resulting subsystems can process in real time at rates far in excess of current digital techniques, with relatively little size, weight, and power. Applications include ELINT, COMINT, RWRs, Laser Radar Doppler Processors, and advanced communications techniques.

### Mathematical Foundations

Consider the Fourier transform S(f) of a signal s(t) which is bandlimited to B and of maximum duration T. The Fourier transform integral is easily rewritten in the form of the “chirp transform algorithm”:

`		S(f)=S(at)=((s(t).R1(t))*R2(t)).R1(t)`
`				    M      C      M`

where . is multiplication, * is convolution, a=B/T is the scale factor, R1(t) is the impulse response of a DDL of length T with chirp rate -a, and R2(t) is the impulse response of a DDL of length 2T with chirp rate a.

The expression for S(at) shows that the sequence of operations on s(t) are: Multiply by a LFM pulse, Convolve by a DDL, Multiply by a LFM pulse. This is the MCM operator.

The dual CMC operator may be derived using the convolution theorem:

`		S(f)=S(-at)=((s(t)*R1(t)).R2(t))*R1(t)`
`				   C       M       C`

### Realization Considerations

POI: The finite duration of R1(t) and R2(t) allow the processing of a T time “frame”. With a system trigger rate of 1/T, the “frames” are contiguous so that a 100% Probability of Intercept (POI) is obtained.

Weighting: For CW input, the processed output is a sinc of -4dB width 1/T, so BT discrete frequencies are resolved over B. For applications with quasi-stationary inputs it is possible and desirable to weight the system to replace the sinc (-13dB sidelobes) output with a Taylor (-35dB sidelobes) output at the cost of a 1.5 resolution reduction. MCM weighting is introduced in time at the first M, and CMC weighting in frequency at the last C.

Multiply: In the unweighted case, the M operators can be realized by doubly balanced mixers with R1(t) and R2(t) as constant level LOs. For weighted MCM, weighting must be introduced thru bilinear multiplication at M1. For weighted CMC with 100% POI, two M operators must be time multiplexed so that concurrent LOs do not produce undesirable intermodulation products.

Multi-Channel Processing: Multiple processor channels may share common M operator LOs, and their DDLs may be thermally stabilized in a common oven. This commonality enhances coherence and tracking accuracy.

From these considerations arise the four common processors described below. They are identified by their operators with any deleted operations enclosed in parentheses:

• MCM: for unweighted 100% POI Fourier transformation applications, weighting requires bilinear multiplication.
• MC(M): for unweighted 100% POI spectrum analysis applications, where the output phase is unused. Multi-channel differential phase is the same as MCM.
• CMC: for weighted 50% POI Fourier transformation applications. 100% POI is achieved by multiplexing two M operators.
• (C)MC: for weighted <=50% POI spectrum analysis applications, where output phase is unused. Multi-channel differential phase is the same as CMC.
 Performance Typical Limit Bandwidth B (MHz) 1-100 500 Time T (us) 1-50 100 BT 100-1000 2500 POI (%) 50-100 100 Resolution -3dB (MHz) .02-1 .01 Weighted resolution -50 dB SQRT (B/T) Weighted sidelobes (dB) 30-35 40-50 Dynamic range (dB) 55-65 70-80 Multi-channel tracking Amplitude (db): 1-2 .5 Phase (deg): 5-10 2.5 Power w/oven (W) 5-30 3-50 Size (cu. in) 10-250 8 IO options: Log video output Real time digital IO w/8 bits I, 8 bits Q

A frequency time-plan for the CMC configuration is shown above. The BT input domain is convolved into the M input domain, mixed with the M LO to obtain the M output domain, and then convolved into the TB output domain. To define the domain boundaries and interior examples, CW signals are represented by solid lines, and impulses by dashed lines. Note that the output domain is simply the input domain with f=(-B/T)t, as stated by the mathematical formulation. For 50% POI simply retrigger (i. e. replicate) the plan at 2T intervals.