### Stretch Operator

We define the *stretch operator* in the *time domain* by

(3.29) |

In other terms, we stretch a sampled signal by the factor by inserting

*zeros*in between each pair of samples of the signal.

In the literature on multirate filter banks (see Chapter 11), the
stretch operator is typically called instead the *upsampling*
operator. That is, stretching a signal by the factor of
is called
upsampling the signal by the factor
. (See §11.1.1 for
the graphical symbol (
) and associated discussion.) The
term ``stretch'' is preferred in this book because ``upsampling''
is easily confused with ``increasing the sampling rate''; resampling a
signal to a higher sampling rate is conceptually implemented by a
stretch operation followed by an ideal lowpass filter which moves the
inserted zeros to their properly interpolated values.

Note that we could also call the stretch operator the *scaling*
operator, to unify the terminology in the discrete-time case with that
of the continuous-time case (§2.4.1 below).

**Next Section:**

Repeat (Scaling) Operator

**Previous Section:**

Power Theorem for the DTFT