If I have a small signal, does routing it through vias to different layer introduce any significant noise on the signal compared to routing it within same layer without any vias? Considering all other parameters like track length, width constant.
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A via doesn't add noise, per se, but it does represent an opportunity for noise to be coupled to your signal that is different from the opportunity along the traces.
Noise couples to your signal through capacitive coupling and inductive coupling. Both will happen along a trace and along a via of any length. How much depends on the aggressor signal, geometries and materials.
NB: As to the other answer indicating that vias create impedance change for high speed digital circuits: This is not really the problem. If you have different trace impedance on different layers, then obviously you will have a reflection from switching layers. Vias can be used in high speed digital circuits to above 25 GBps as described in my blog here.
A via doesn't add noise, per se, but it does represent an impedance change that can create extra reflections of high-speed signal edges. This is why they're avoided in the traces used for things like DDR SDRAM, high-speed USB and Ethernet connections.