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# Properties of Z-Transform

The z-transform has a set of properties in parallel with that of the Fourier transform (and Laplace transform). The difference is that we need to pay special attention to the ROCs. In the following, we always assume

and

• Linearity

While it is obvious that the ROC of the linear combination of and should be the intersection of the their individual ROCs in which both and exist, note that in some cases the ROC of the linear combination could be larger than . For example, for both and , the ROC is , but the ROC of their difference is the entire z-plane.

• Time Shifting

Proof:

Define , we have and

The new ROC is the same as the old one except the possible addition/deletion of the origin or infinity as the shift may change the duration of the signal.

• Time Expansion (Scaling)

The discrete signal cannot be continuously scaled in time as has to be an integer (for a non-integer is zero). Therefore is defined as

Example: If is ramp

 1 2 3 4 5 6 1 2 3 4 5 6

then the expanded version is

 1 2 3 4 5 6 0.5 1 1.5 2 2.5 3 1 2 3 0 1 0 2 0 3

where is the integer part of .

Proof: The z-transform of such an expanded signal is

Note that the change of the summation index from to has no effect as the terms skipped are all zeros.

• Convolution

The ROC of the convolution could be larger than the intersection of and , due to the possible pole-zero cancellation caused by the convolution.

• Time Difference

Proof:

Note that due to the additional zero and pole , the resulting ROC is the same as except the possible deletion of caused by the added pole and/or addition of caused by the added zero which may cancel an existing pole.

• Time Accumulation

Proof: The accumulation of can be written as its convolution with :

Applying the convolution property, we get

as .

• Time Reversal

Proof:

where .

• Scaling in Z-domain

Proof:

In particular, if , the above becomes

The multiplication by to corresponds to a rotation by angle in the z-plane, i.e., a frequency shift by . The rotation is either clockwise () or counter clockwise () corresponding to, respectively, either a left-shift or a right shift in frequency domain. The property is essentially the same as the frequency shifting property of discrete Fourier transform.

• Conjugation

Proof: Complex conjugate of the z-transform of is

Replacing by , we get the desired result.

• Differentiation in z-Domain

Proof:

i.e.,

Example: Taking derivative with respect to of the right side of

we get

Due to the property of differentiation in z-domain, we have

Note that for a different ROC , we have

Next: Z-Transform of Typical Signals Up: Z_Transform Previous: Properties of ROC
Ruye Wang 2014-10-28