![]() There's a crucial practical difference, in that we literally perform Discrete Fourier Transforms on concrete input vectors to produce concrete output vectors. The z-transform equation is closely related to that for the DFT. The expression for the system eigenvalues in terms of z is known as the "z transform" of h (for n from -infinity to infinity): The product of the input coefficient \(a_k\) and the system's eigenvalue Of complex exponential sequences, with each coefficient being That is, the output is also a linear combination of the same set Then the system output for an LTI system with impulse response h will be This "eigenrelationship" makes it convenient to express a signalĪs a linear combination of complex exponentials. \(H(z)\) as the eigenvalue, since \(M_h\) times \(z^n \)equals the constant \(H(z)\) times \(z^n\).īecause of the superposition property of linear systems, ![]() We can see that the complex exponential \(z^n\) is an eigenvector of \(M_h\), with Response h in its rows, as discussed in an earlier that is, a matrix with a set of shifted time-reversed copies of the impulse Response h in the more general form of the "convolution matrix" \(M_h\) We can see another way to say this if we express the LTI system with impulse Then we can rewrite the system equation as On the value of z and on the impulse response h. The output will be the convolution sum h # x, orĪnd the output is \(z^n\) multiplied by a constant that depends Has as its input one of these exponential sequencesįor some (arbitrarily chosen) complex number z. Now suppose a linear time-invariant (LTI) system with impulse response h(n) Spiral like those we have just seen, whose magnitude is exponentiallyĭecreasing or increasing depending on whether z has a magnitude less than Is slightly less than or greater than 1, then z^n will be a sinusoidal If z is a complex number (with a non-zero imaginary part) whose magnitude So that z^n will be e^(i*n*w), or cos(n*w) i*sin(n*w). This can easily be shown by considering that in this case z = e^(i*w)įor some real number w, which is equivalent to cos(w) i*sin(w), ![]() Sinusoid sampled at intervals of angle(z) radians. The y-axis representing the imaginary part), then z^n will be a The complex plane, with the x-axis representing the real part of z and If z is a complex number that happens to lie on the unit circle (in Increase exponentially, remain constant, or decrease exponentially in Z is less than, equal to, or greater than -1, these values will If the number z is a negative real number, we will get a sequence thatĪlternates between positive and negative values. If z is a positive real number, we will get a sampledĮxponential ramp, that is either rising or falling depending on whether If the number z happens to be one or zero, we will get a sequence You should start with a clear graphical intuition about what such sequencesĪre like. X = z^n for n from -infinity to infinity, z some complex number Made up of integer powers of some complex number: In this segment, we will be dealing with the properties of sequences Introduction to the z transform INTRODUCING THE Z-TRANSFORM Background
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