What is the digital down converter in an SDR and how does it extract a narrowband signal from a wideband digitized spectrum?

The digital down converter (DDC) in an SDR is a digital signal processing function (typically implemented in an FPGA) that extracts a narrowband signal of interest from the wideband digitized spectrum captured by the ADC. The DDC performs three essential operations: frequency translation (mixing the digital samples with a numerically controlled oscillator, NCO, to shift the desired signal from its RF or IF frequency to baseband or near-DC), lowpass filtering (applying a digital filter to reject all signals outside the desired channel bandwidth), and decimation (reducing the sample rate to match the narrowband signal's Nyquist requirement, reducing the data rate that must be transferred and processed by the host). For example, an ADC sampling at 200 MSa/s digitizes a 100 MHz wide spectrum. A DDC tuned to extract a 25 kHz wide AM signal at 50 MHz IF would: multiply by a 50 MHz NCO (shifting the signal to 0 Hz), apply a digital lowpass filter with 25 kHz passband, and decimate by 4000 (reducing the output rate from 200 MSa/s to 50 kSa/s). The output is a complex (I/Q) baseband signal containing only the desired 25 kHz channel, at a data rate 4,000 times lower than the ADC output. This massive data rate reduction is what makes it practical to stream the selected channel to a host processor over standard interfaces.