# Slow and Fast Fading in Wireless Channels

We discussed the idea of fading in wireless channels in a previous article. To understand different types of fading in the context of time variations, refer to the figure below that shows a multipath channel.

A slow motion scenario is illustrated in the figure below where three multipath components are arriving with Doppler shifts $F_{D,i}$ from the carrier frequency. In this scenario, the magnitudes of $F_{D,i}$ are small and hence observe very little spreading of the cumulative spectrum.

This can be understood by recalling that when two sinusoids with two different frequencies $F_1$ and $F_2$ are added, the resulting signal contains both a sum frequency $F_2+F_1$ and a difference frequency $F_2-F_1$. The envelope of the signal arising from this summation of two sinusoids is given by the difference frequency $F_2-F_1$. When this difference is small, it takes a long time for the envelope to go from one zero to the next and the channel amplitude can be taken as a constant for a large number of symbols. The above figure reveals that the signal bandwidth $B_{\text{QAM}}$ plays a central role in determining how much effect the Doppler spread has on the Rx output. As long as the largest Doppler frequency difference is still a fraction of the signal bandwidth,
\begin{equation*}
|F_{D,\text{max}}-F_{D,\text{min}}| ~<<~ B_{\text{QAM}} \end{equation*} as illustrated in the above figure, the motion effects are minimal and the signal undergoes time flat or slow fading. In time domain, a channel is time flat if it is changing very slowly with respect to the symbol rate $1/T_M$, i.e., the coherence time is much larger than the symbol time $T_M$. As a consequence, the Rx sees an almost constant channel response without any significant variations for a large number of symbols as illustrated in the figure below.

Next, we explore the topic of fast fading.

A fast motion scenario is illustrated in the figure below where three multipath components are arriving with Doppler shifts $F_{D,i}$ around the carrier frequency. In this scenario, the magnitudes of $F_{D,i}$ are large causing a significant spreading of the cumulative spectrum.
Again, the difference between the frequencies of two sinusoids forms the envelope. When this difference is large, the signal envelope exhibiting a short period quickly complete its repetitions and the channel amplitude varies in proportion, even within a symbol time $T_M$ in some cases. We say that the fading rate is fast or time selective.
Notice that the signal envelope, being inversely proportional to the significant Doppler spread, either changes within a symbol time $T_M$ or very few symbols experience the same fade as illustrated in the figure below. We say that the coherence time is less or on the order of a symbol time $T_M$.