First of all, we observe that this wave packet consists of a real amplitude, a 'sinc' function, multiplied by an exponential phase factor, which is rapidly oscillating when the integer is large. From the viewpoint of engineering one says that the wave train is getting modulated by the `sinc' function. The resultant wave train amplitude has its maximum at . From the viewpoint of physics one says that the wave trains comprising the wave packet exhibit a beating phenomenon with the result that they interfere constructively at . From the viewpoint of mathematics one observes that the integral has a maximum value when the integrand does not oscillate, i.e. when .
Second, we observe that the spacing between successive zeroes is . They are located at
At the wave packet has maximum modulus . These two properties are summarized by the sifting property of :
Third, it has mean frequency . Its mean position along the time axis is . Its frequency spread is the width of its frequency window in the Fourier domain
Its temporal spread,
is its half width centered around its maximum, which is located at . Consequently, the frequency spread times the temporal spread of each wave packet is
which is never zero. Thus, the only way one can increase the temporal resolution is at the expense of the frequency resolution, i.e., by increasing the frequency bandwidth of each wave packet. Conversely, the only way to increase the frequency resolution is to increase the width of the wave packet.
The last property is expressed by the following exercise:
where is a fixed positive constant.
The completeness relation, Eq.(2.71) is equivalent to the statement that any square integrable function can be represented as a superposition of wave packets, namely
are the expansion coefficients.