It's not impossible to overcome -- in fact, it's a lot easier to fix the problem at the source than it is to fix it later in post-production.
Let's look at a couple of the most common problems that will plague your home studio (or live performance setup), and how you can easily correct them. Even on a limited budget, these problems aren't hard to fix; fixing them will make a huge difference in your end product.
Buzz & HumBuzz and hum are deeply annoying; they're also very common, depending on the balancingof the cables in your studio. There's two potential sources for these noises, but they're also the easiest to correct, especially on a limited budget.
First is buzz caused by power grounding. Hum caused by power is always centered at right around 60Hz due to the 120Hz oscillation of power coming into your wall (in the United States, anyway; in other countries, the power standards are different).
The best way to deal with a single item that's humming due to grounding issues is to use a ground lift adapter; this is also commonly known as a "three prong adapter", because they're commonly used to adapt a grounded connector to an ungrounded outlet.
Using a ground lift adapter is my go-to method for fixing hum and buzz in the studio, as well as live.
Aside from this, if you're getting buzz on a channel through a direct box (DI box), many times you'll see a "ground lift" switch. This does the same thing as using a plug adapter; it'll promptly put an end to the buzzing (or, worst case, make it less noticeable).
Radio (RF) InterferenceIn my old apartment, I had massive radio frequency interference, or "RF", problems. RF problems can be one of the most frustrating things you've ever dealt with.
RF interference happens when, due to a shielding issue in your wall or in your cabling, the entire power or signal path becomes an antenna.
RF ("radio frequency") interferences are hard to diagnose, because the sources for that interference may chance frequently. Any aspect of the system, including cabling and internal wiring, is susceptible to RF noise. Mainly, it's likely to be your power cables acting as antennas for strong AM stations nearby, and it's especially noticeable if it's your monitors making the noise.
Speaker drivers in your monitors are usually the most obvious victim, due to their ability to make anything loud; you'll be hearing radio stations in the middle of your mix session, or a preamp will pick up private radio communications.
True story: a friend's studio was next door to a cab dispatch company in south Miami; he enjoyed hearing a significant portion of his mixing sessions interrupted by cabs being dispatched constantly.
If you can't diagnose the problem with cabling, it's time for some ferrite chokes. These small devices contain a small amount of ferrite in a circular form, which can be taken apart and wrapped around any cable you see fit.
These chokes are small and inexpensive, and can be placed inconspicuously in your signal chain. These clip over your power or input cables, and keep your signal clean without any massive re-wiring necessary. With a ferrite choke in place, you've got the potential to fully eliminate RF noise. You might, however, have to experiment with positioning the choke on the line.
Power ConditioningThere's much debate over power conditioning for a home studio. Quite simply, a power conditioner makes sure that the power entering your equipment is clean, free of noise, and at a steady voltage. This is important because it can not only improve the quality of a recording, but it protects your equipment, as well.
Power spikes are one of the most common reason that perfectly good equipment is electrically "bricked" -- made useless due to being "fried" due to a power voltage issue.
Some people say that power conditioners offer no additional protection or added benefit to sound quality; in my experience, especially in apartments, power conditioning serves a very valid purpose.
Good power conditioners aren't cheap; at minimum, you'll want to buy one with multiple outlets and a user-replaceable safety fuse. Many of the top brands will also offer an insurance policy on your gear; if your gear is damaged by a power spike while under the protection of the conditioner, you'll receive a replacement. Remember, many of the cheaper "conditioners" aren't much more than a glorified power strip.
Noise, hum, and buzz certainly are frustrating -- but armed with a little knowledge, you'll be able to save the day (and, possibly, save an important session)!