I've read that OFDM normally uses a one-tap equalizer. It seems that a one-tap equalizer could only scale and delay a signal. Why is this useful?
I thought multipath was handled through the cyclic-prefix and by choosing a symbol width that is large compared to the delay spread. Why is an equalizer needed at all?
The OFDM signal as a whole is affected by frequency selective filtering. It is usually designed such that the subcarrier bandwidth is smaller than the channel coherence bandwidth. This yields you frequency flat fading for each subcarrier which can be described by a single complex multiplication and equalization can equally done with just one tap.
Equalization in OFDM usually happens in the frequency domain. Let $X$ be the vector of subcarriers in the frequency domain and $x=IDFT(X)$ the corresponding time domain signal. Then, if the impulse response of your channel is a vector $h$, the received signal $y$ will be $y=x*h$ where $*$ referes to convolution. Then $Y=DFT(Y)=X \cdot H$ with $H=DFT(h)$ and $\cdot$ refering to element-wise multiplication. However, this is only true if $y$ is the circular convolution of $x$ and $h$ while in reality the convolution is linear. Therefore, the cyclic prefix is prepended (it needs to be at least as long as $h$) so that the linear convolution contains the circular convolution.
Then, $Y=DFT(Y)=X \cdot H$ with $H=DFT(h)$ and $\cdot$ as stated before. Since each element of $Y$ and $X$ refers to one subcarrier it can be seen that each subcarrier is multiplied with just one complex element from $H$ and hence equalization just becomes $Z=Y/H$ where again $/$ referes to element-wise division. This describes zero-forcing equalization which might not be the best choice if additive noise and interferers are present but it transports the idea.
And this is nothing else but one-tap equalization for each individual subcarrier.
Multipath propagation of the transmitted OFDM symbol will exhibit as frequency selective fading across the frequency spectrum of the received OFDM symbol. This means different subcarriers will undergo different gains due to superposition of delayed and scaled transmitted OFDM symbols at the receiver. When observed across the symbol in frequency, this is not frequency flat. However, for one subcarrier (infact subcarrier spacing) this is constant given subcarrier spacing is kept lesser than inverse of maxium delay spread. Inverse of maximum delay spread is called coherence bandwidth. One tap equalizer is a term used for equalization of one subcarrier or just one bin of the OFDM frequency domain symbol at the receiver. One tap equalizer can be zero forcing or MMSE and MMSE outperforms zero forcing equalizer.
