Receiver and waveform mathematics for thermal noise, noise figure, bandwidth, ADC performance, integration, detection thresholds and false alarms.
A radar or EW receiver is not limited only by antenna signal strength. Once energy reaches the front end, it competes with thermal noise, receiver noise figure, interference, distortion, phase noise, quantization and detection thresholds. A strong waveform at the antenna can become a weak measurement if the receiver chain wastes dynamic range or admits too much noise bandwidth.
For EW, signal math is also classification math. Pulse width, PRI, frequency, modulation, amplitude, phase and time-of-arrival quality determine whether an emitter can be detected, identified, geolocated and tracked.
Analog waveform, quantized samples and noise-like scatter: SQNR, SINAD and ENOB describe how much measurement quality survives digitization.
The receiver chain turns RF energy into measurements. Every stage can improve selectivity or damage margin. The noise floor is the baseline power against which weak signals must be detected.
Formula reading: N is thermal noise power in watts. k is Boltzmann's constant, T is physical noise temperature in kelvin, and B is receiver noise bandwidth in hertz. At room temperature, thermal noise density is about -174 dBm/Hz. Increasing bandwidth by a factor of two raises noise by 3 dB. NFdB is receiver noise figure, added because a real receiver contributes internal noise.
Noise figure measures how much the receiver degrades input SNR. Sensitivity defines the weakest useful signal for a required output SNR or probability of detection. Dynamic range defines how much weak and strong signal content can coexist without compression or masking.
Formula reading: F is linear noise factor. If F = 2, the receiver halves the SNR, which is 3 dB noise figure. Te is equivalent input noise temperature: the temperature of an ideal input noise source that would create the same extra noise as the receiver. Lower NF directly improves sensitivity.
Formula reading: Smin is receiver sensitivity for a required SNR. B is bandwidth in hertz. SNRreq is the threshold needed by the detector or demodulator. DR is dynamic range between the weakest useful signal and the front-end compression limit P1dB. In EW, insufficient dynamic range causes strong emitters to hide weak emitters.
Digital receivers sample amplitude in time and quantize it into finite levels. More bits help, but real ADC quality is limited by noise, distortion, clock jitter and analog front-end linearity.
Formula reading: N is ideal ADC bit count. Each bit adds about 6 dB of ideal quantization dynamic range. SINAD includes noise and distortion, so ENOB is the number of effective bits actually achieved. fs is sampling rate and must exceed twice the sampled bandwidth to avoid aliasing, before practical filter guard bands are considered.
A radar rarely decides from one sample. It integrates pulses, filters waveforms and compares cells against adaptive thresholds. Integration can improve SNR, but target fluctuation, phase stability and clutter determine how much gain is actually usable.
Formula reading: Gint is ideal noncoherent/incoherent-style integration gain estimate from Np pulses. Tdwell is time-on-target for a scanning beam. FAR is false alarm rate. Raising threshold lowers false alarms but increases required SNR; lowering threshold improves weak-target sensitivity but can overload the tracker or operator with false detections.
An ESM receiver estimates parameters that become emitter identity: carrier frequency, pulse width, time of arrival, pulse repetition interval, bandwidth, modulation type, amplitude, angle of arrival and scan pattern. Measurement accuracy depends on SNR, bandwidth, sampling clock stability and receiver calibration.
Formula reading: Δt is timing uncertainty. In radar range, timing uncertainty maps to range uncertainty through c0Δt/2. Frequency resolution improves with observation time Tobs; a longer coherent observation can separate closer frequencies, assuming the signal remains stable enough.
A pulse descriptor word stores measured pulse parameters such as TOA, RF, PW, amplitude and AOA. Many EW functions are built from sequences of these words: deinterleaving, PRI analysis, scan analysis, emitter identification and threat prioritization.
Once pulses are detected, the receiver creates pulse descriptor words and the ELINT processor tries to decide which pulses belong to which emitter. That pulse sorting step is deinterleaving.
Detection answers: did a pulse occur? Measurement answers: what are its TOA, RF, PW, amplitude and AOA? Deinterleaving answers: which emitter produced it? The dedicated module covers PRI histograms, pulse-train hypotheses, clustering, stagger, jitter, agile emitters and track confidence.