# Tuning a radio to a certain frequency

I've a fundamental problem with understanding terminology used in DSP. What does it mean to tune a radio (for example, a software defined radio) to a certain carrier frequency $f$? The received signal is an arbitrary waveform and not a sine wave of frequency $f$. I understand AM modulation: you take a sine and manipulate its amplitude. The FM modulation I don't get: the modulated signal is not a sine, so why do we say that we tune a radio to a certain frequency in that case?

• I think tuning here means setting the transmit or receive frequencies to the frequency you would like to transmit or receive the data. If you have transmit at a frequency say 50 Mhz, you need to set your receive frequency in the radio to 50 Mhz,otherwise you cannot receive the signal sent. Your question is unclear. please edit it to get a better response. – Karan Talasila Jul 4 '14 at 1:34
• Short answer: modulated signals are modulated atop some sinusoidal carrier frequency that corresponds to the center of the signal bandwidth. When you tune a radio to a particular carrier frequency $f$, you're selecting which portion of the spectrum that the receiver will observe; it doesn't mean that you're only looking at the content at the carrier frequency. – Jason R Jul 4 '14 at 3:58
• @JasonR: so tuning to a frequency $f$ really means tuning to a band of frequencies centered at $f$? The radio observes a band of sinusoids and composes them to a waveform? – aaa Jul 4 '14 at 7:54
• As JasonR commented, the transmitted signal is modulated by a carrier frequency $f$ (i.e. shifted in the frequency domain by $f$) If you are interested, look here. Tuning the radio to frequency $f$, means filtering the signal to select the correct portion of the spectrum and demodulating the signal back to its original frequency content (i.e. shifting the signal in the frequency domain by $-f$). – ThP Jul 4 '14 at 9:16
• @ThP Shifting in frequency domain by $f$ means adding the signal to $f$? – aaa Jul 4 '14 at 11:22

## 2 Answers

there are a lot of misconceptions in the question.

"The received signal is an arbitrary waveform and not [only] a sine wave of frequency $f$."

it's a collection of sine waves at different frequencies. and "tuning" a radio or a filter or whatever is selecting frequency components in the neighborhood of $f$ while rejecting the others.

A mathematical theorem (named after Fourier) says that (non-pathological) signals that are "not a sine" are, for all practical purposes, identical to (or can be accurately described as) the sum of a bunch of pure sinusoids (of various phases and magnitudes).

As practical radios don't have infinitely narrow bandwidths, when tuning to a single pure frequency, you actually get a suitable range of frequencies corresponding to a bunch of pure sinusoids, enough to reasonably describe and thus demodulate your "not a sine" original signal.