Yes, it is true, modern audio specrum analyzer uses DSP to extract the spectral component of the audio signal.
However both Analog Devices and Texas Instruments make available very low cost ev-board of their device and the application you want to build is very common and so I think you can easily find the algorithm you need (substantially an FFT)
In the zip file attached you will find several audio algorithms for ADSP2181 with the m code for simulate them on matlab.