ADSP Linux Kernel for release 3.1.2 based on Linux 5.15.168.
Minimal requirements to compile Kernel https://docs.kernel.org/process/changes.html
This steps applicable for all supported platforms and this is not complete guide for kernel development.
There is plenty resource on kernel development online:

https://kernelnewbies.org/KernelBuild
https://linux-kernel-labs.github.io/refs/heads/master/labs/arm_kernel_development.html
https://www.raspberrypi.com/documentation/computers/linux_kernel.html

Obtain kernel sourcecode

git clone https://github.com/analogdevicesinc/lnxdsp-linux.git

default configuration for SC598

ARCH=arm64 CROSS_COMPILE=aarch64-linux-gnu-  make sc598-som-ezkit_defconfig

default configuration for SC589-mini

ARCH=arm CROSS_COMPILE=arm-linux-gnueabi- make sc589-mini_defconfig

compile kernel device tree and kernel image for SC598

ARCH=arm64 CROSS_COMPILE=aarch64-linux-gnu- make -j`nproc` Image.gz modules dtbs

compile kernel device tree and kernel image for SC589

ARCH=arm CROSS_COMPILE=arm-linux-gnueabi- make -j`nproc` zImage modules dtbs

Currently, the default boot format used is fitImage. Here is shared example how to rebuild fitImage with newly compiled kernel.

mkdir new_fit_image
cd new_fit_image
cp <buildir>/tmp/deploy/images/adsp-sc598-som-ezkit/fit-image.its ./
cp <buildir>/tmp/deploy/images/adsp-sc598-som-ezkit/adsp-sc5xx-ramdisk-adsp-sc598-som-ezkit.cpio.gz ./
cp <kernel_buildir>/arch/arm64/boot/dts/adi/sc598-som-ezkit.dtb ./
cp <kernel_buildir>/arch/arm64/boot/Image.gz ./
mkimage -f fit-image.its fitImage

boot fitImage with your preferred boot method.

• kernel modules compiled need to be copied to filesystem in use, if NFS boot is used copy NFS boot directory
• after changes/modifications to kernel is done create a patch and add to your yocto layer or create custom repository and point recipe to that
• apply required patches from meta-adi Yocto layer kernel recipe