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Z Transforms

Z Transforms. Dr. Veton Kepuska. z-Transform.

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Z Transforms

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  1. Z Transforms Dr. Veton Kepuska

  2. z-Transform • In mathematics and signal processing, the Z-transform converts a discrete-time signal, which is a sequence of real or complex numbers, into a complex frequency domain representation. It can be considered as a discrete-time equivalent of the Laplace transform. This similarity is explored in the theory of time scale calculus. • The z-transform is useful for the manipulation of discrete data sequences and has acquired a new significance in the formulation and analysis of discrete-time systems.  It is used extensively today in the areas of applied mathematics, digital signal processing, control theory, population science, economics.  These discrete models are solved with difference equations in a manner that is analogous to solving continuous models with differential equations.  The role played by the z-transform in the solution of difference equations corresponds to that played by the Laplace transforms in the solution of differential equations.

  3. z-Transform • Definition: Consider a function x(t) defined for t≥0. As we should know we could sample this continuous function at times t = T, 2T, 3T, … where T is the sampling period (or sampling rate). We can write the sample as a sequence using the notation • Given a finite length sequence x[n], defended in the interval [0,N] and z-any complex number, the z-transform is defined as:

  4. z-Transform • This transformation produces a new representation of x[n] denoted by X(z). • Returning to the original sequence (via inverse z-transform) x[n] requires finding the coefficients associated with nth power of z-1.

  5. z-Transform • Formally transforming from the time/sequence/n-domain to z domain is represented as:

  6. z-Transform • A sequence and its z-transform are said to form a z-transform pair: • n is operator in the domain of sequence indicating the sample is is considered an independent variable. • In the z domain the independent variable is z.

  7. Example 1 • Appling the definition of the z transform:

  8. Example 2 • Appling the definition of the z transform:

  9. Example 3 Appling the definition of the z transform:

  10. The Z transform and Linear Systems

  11. The Z transform and Linear Systems • The z-transform is specifically useful in the analysis and design of LTI systems

  12. FIR Filter • A system that has only for k=1,…,N is said to be a Finite Impulse Response (FIR) filter. • The name reflects the fact that FIR filters have finite impulse (e.g., unit sample) response. • FIR filters are also called moving average (MA) filters considering the fact their output is simply a weighted average of the input values.

  13. The z-Transform of an FIR Filter • We should recall that for any LTI system with input x[n] and impulse response h[n], the output y[n] is: • We are interested in the z-transform of h[n], where h[n] is an FIR filter

  14. Consider the input: • The output y[n] is:

  15. The term under the parenthesis is the z-transform of h[n] • This is also termed the system function • This function is defined as:

  16. The z-transform pair that was just established is

  17. System Function is an Mth degree polynomial in complex variable z. • As with any polynomial, it will have M roots or zeros, that is there are M values, z0, such that H(z0) = 0; • These M zeros completely define the polynomial, that is: Where zk, k=1,…,M denote zeros of polynomial.

  18. Example 4 • Find the zeros of: • The z-transform is

  19. The zeros of H(z) are -1/2 and +1/3 • The difference equation has the same zeros, but a different scale factor;

  20. Properties of the z-Transform

  21. Properties of the z-Transform • The z-Transform has a few very useful properties, and is defintio extends to infinite signals/impulse responses (IIR).

  22. The Superposition (Linearity) Property • The Superposition (Linearity) Property

  23. Proof:

  24. The Time-Delay Property Proof:

  25. The Time-Delay Property Let Hence

  26. Similarly

  27. A General z-Transform Formula • We have seen that for a sequence x[n] defined over the interval 0 ≤ n ≤ N the z-transform is This definition extends for sequences having interval from -∞ ≤ n ≤ ∞

  28. Properties of z-TranSform • z-Transform Convolution Theorem: • Cascading Systems LT 2 H2(z), h2[n] LT 1 H1(z), h1[n] y[n] Y(z) x[n] X(z) w[n] W(z)

  29. The Z-Transform as an Operator

  30. The Z-Transform as an Operator • The z-transform can be considered as an operator. Unit-Delay Operator UnitDelay x[n] y[n]= x[n-1] z-1 x[n] y[n]= x[n-1]

  31. Unit DelAy Operator • In the case of the unit delay, we observe that: Which is derived from the fact that Y(z) = z-1X(z)

  32. The Z-Transform as an Operator • The filter described with the following equation: This expression can be viewed as the operator: This is so because:

  33. Example: Two-Tap FIR Filter x[n] X(z) z-1 b0 z-1X(z) b1 x[n-1] y[n]

  34. Example: Two-Tap FIR Filter • Using the operator convention, we can write by inspection:

  35. FIR Filters

  36. FIR Filters • General constant-coefficients equation: • Impulse (unit sample) response, h[n], of the filter is obtained when x[n]=δ[n].

  37. Linear Phase of FIR Filters • If FIR filter has coefficients that are symmetric, as depicted in the following relationship: • Then, it can be shown that the resulting filter has linear phase⇒constant delayfor all frequency components of the input signal. • This property is very important in many communications data streams (speech, data, …) and image processing applications.

  38. FIR Filter Structures • For a given difference equation there are different ways to implement a digital filter. • Selection of a particular filter structure to be implemented is dependent on many factors: • Programming considerations • Hardware • Sensitivity of Quantizing Coefficients • Quantization noise of the input signal.

  39. Direct Structure of an FIR filter x[n-1] x[n-2] x[n-M] D D D x[n] βM β2 β0 β1 y[n]

  40. Symmetric – Linear Phase FIR filter β0 y[n] • Better Implementation: x[n] D D β1 x[n-1] D D βL/2 x[n-2] D x[n-M/2+1] x[n-M/2]

  41. MATLAB • >> filterDesigner

  42. Filter Designer • Equiripple • Least-Squares • Window • Constrained Least-Squares • Complex Equiripple • Maximally Flat • Least P’th Norm • Constrained Equiripple • Generalized Equiripple • Const. Band Equiripple • Interpolated FIR

  43. Windowing Method • Choose Chebyshev Filter

  44. Chebyshev Window Method

  45. Convolution and the z-Transform

  46. Convolution and the z-Transform • The impulse response of the unity delay system is: • The system output written in terms of a convolution is: • The system function (z-transform of h[n]) is:

  47. Convolution and the z-Transform • Hence in general, when the system is described by H(z) we can optain the output Y(z) knowing what the input is X(z) To show this we start with:

  48. Convolution and the z-Transform

  49. Example: Convolving a Finite Duration Sequences • Suppose that: Find first Y(z) by applying z transform to both x[n] and h[n] Then we find the Y(z) by applying the multiplication of z transform rule:

  50. Example: Convolving a Finite Duration Sequences Then we find y[n] by inverse z transformation:

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