Expected inflation in the real interest rate equation Real interest rate = Nominal rate - Expected inflation

In the above equation, in a quarterly data-set, which expected inflation shall be used? next quarter (q+1) or the same quarter of next year (q+4)? and why?
Your formula isn't quite right. If $r$ is the nominal interest rate, and $\pi$ is the inflation rate, the real interest rate is $\frac {1+r}{1+\pi}-1$. The formula $r-\pi$ is approximation for small rates. (Note: for all of this, I'm using &quot;rate&quot; to mean the multiplicative factor minus one. So if the balance is multiplied by 1.2, the rate is 0.2.)
The inflation should be over the period over which the interest rate is being calculated. It's like to like: the real interest rate at a particular time is given by the nominal interest rate at that time compared to the inflation at that time.
The fact that you think you are wondering whether you should use the future inflation suggests that you may be confusing the spot inflation rate with the Consumer Price Index. You can get the real interest rate by taking the total nominal interest times the ratio between the CPI at the end of the period to the CPI at the beginning. That is, if $r$ is the total nominal interest, $CPI_i$ is the starting CPI, and $CPI_f$ is the final CPI, then the total real interest is $(1+r) \frac {CPI_f}{CPI_i}-1$.