Math test

Prompt

A plugin claims zero latency and unity gain. You can’t see its code. A null test is run on it: output minus input; no host latency compensation; steady sine tones, at 48 kHz. These are the residuals, relative to the input level: Frequency Residual 100 Hz −38.5 dB 1 kHz −23.6 dB 10 kHz −3.9 dB Assume the plugin behaves as a constant gain plus a constant delay, which may be fractional. What is the delay, in samples? What is the gain, in dB? The developer updates the plugin to report 1 sample of latency, and changes nothing else. In a host that compensates for reported latency, what is the 10 kHz null residual now? What is the smallest change that gets the null below −35 dB at every frequency? Be explicit about anything the data cannot determine. Answer each in one or two lines.