How can you implement a DSP system on a tight budget?

This can also be accomplished with an RPi and a HAT or USB ADC and/or DAC, depending on your needs. This would allow for advanced FIR processing at high levels. Another cheap alternative is ADI based development boards that use sigmastudio and ADAU1452-based chips (replacing the older ADAU1701-based boards). Once you are beyond the learning curve of the software, you can create limiters (short-term and general), perform delay alignments, volume leveling, PEQ IIR functions, some FIR taps for phase alignment, among many other functions. If you prefer to enter the filters for PEQ as biquads, that can be done. Just to give some alternatives.

Issues with using the Raspberry Pi are cost of high-quality DACs as HATs, limit of ADCs and digital input boards to receive signals from external sources, and limits due to USB based DACs depending on number of channels needed for the specific setup. The RPi can perform all the FIR processing you need, but there are limitations on implementations that need thought through before adopting this solution as a high-end audio processing solution. That is not to say you should not do it, as this can be a great solution. But there are costs that need considered and planning is needed for the signal chain before you begin.

Not always will I point to cheap development boards, but the ADAU1452 boards on AliExpress can be a cost effective entry into the realm of DSP. More capable than the ADAU1701 boards, cheaper than the RPi after adding in the cost of appropriate quality DACs and possible digital audio inputs or HATs with ADCs, these can be a great resource to add to your arsenal. With the SHARC based processors moving toward other programs and modular interfaces, the sigmastudio based designs are diverging greatly from SHARC, but you can still design your own algorithms for implementation if you have a little know-how. And Analog Devices forum has been a great help on past projects.