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This is a necessary evil to get the mathematical description of the sampled system. The transfer function must be the Digital Signal Processing system description, since by the preceding definition, if you multiply it by the z-transform of any input signal, you get the z-transform of the output signal as shown hereafter, with capitals representing the z-transforms of the signals.
In the equation above, x t is the input into the ADC and y t is the output from the DAC and T z must be the DSP system description, since if you multiply it by the z-transform of the input, you get the z-transform of the output. The reason for using z-transforms is that the preceding equation is valid. If you took the ratio of sampled output signal to sampled input signal to a DSP system, you would get some expression, but it would change for each input signal.
By using the z-transforms of the signals, the ratio of any output to the corresponding input is always the same expression and must be the system math description, since when it is multiplied by the input, you get the output. With a math description of a digital filter, it is then possible to analyze its characteristics and even design a filter to meet the desired graphical specifications shown in Chapter 3. The definition of the z-transform of a sampled signal f n is F z , as defined by Equation 4.
If the signal out of an ADC consists of a few samples, the z-transform of this signal, F z , is easy to get, as shown in the Example 4. One important property of a z-transformed signal using Equation 4. This is a good thing to check. If an equation has been z-transformed and the sample number n appears anywhere in the equation, a mistake has been made! Example 4. The z-transform of a signal with only a few sample values Problem: Let the input signal f t into an ADC be 2t for t greater than zero and less than 4, and zero otherwise.
Using the definition for the z-transform of a signal in Equation 4. The z-transform of the impulse allows the discrete time equation of a few samples to be written mathematically and then the z-transform can be taken, as shown in Example 4. One of the major uses of the sampled unit step is to start and stop a signal. This is shown in one of the problems at the end of the chapter.
In the preceding section, we determined the z-transforms of two basic sampled signals, the unit impulse and the sampled unit step. In this section we expand on the pairs of sampled signals and the corresponding z-transform, using basic algebra. It is useful also to relate the sampled signal coming out of an ADC to the analog signal coming in, which is shown in column 1 of Table 4. The definition of the z-transform of a sampled signal shows that if any sampled signal is A times bigger, then its z-transform is A times bigger, as shown in the following equation.
This signal and its sampled equation are given in the following equations. The z-transform just given is one of the most important in DSP. Later it will be used to determine the stability and other properties of any DSP system. Table 4. The mathematical derivation of some of these is done the same way as that for the exponentially decaying sampled signal, but the final closed form solution requires the use of the Euler equation, which is not given until Chapter 6.
The organization of Table 4. The z-transform of these sampled signals is not the z-transform of the product of the z-transforms of each component of the product. The z-transform was obtained by the same methods shown for the decaying exponential sampled signal The z-transform of the product of two signals is not the product of the z-transforms. Using Table 4.
Thus the sampled signal and the z-transform of the sampled signal are given in the following equations. Using basic algebra we will use the definition of the z-transform to derive the major property of z-transforms.
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Later this property will allow the student to go back and forth between the DSP system math description T z in terms of the variable z and the difference equation given in Chapter 1, which is the equation that is actually coded. This property will be developed in two different ways for the student. This property is called the shifting property, and it relates the z-transform of sampled signals that are time-shifted versions of each other if the time shift is in integer multiples of the sample time T. Remember that any signal or function of time that is delayed by kT is written in terms of the unshifted signal f nT as f nT — kT , or f n — k.
After the first method of derivation of this property, examples using it will be given. If you have the representation of the z-transform of a sampled signal f n either as F z or as a specific function of z, it is very helpful to use this to write the time-shifted sampled signal, since the difference equation in Chapter 1 is in terms of shifted and unshifted input and output sampled signals.
This representation is derived next. From the definition in Equation 4. The significance of the preceding equation called the shifting property is that if the differenc e equation for a digital filter is given, the use of this property on the equation gives an algebraic equation for the digital filter. Another significant property is shown in later chapters when we design a digital filter or DSP system, which is mathematically expressed in terms of the variable z, we can use the property to get the difference equation to code in a computer or a DSP chip.
In the next chapter this property is used on the DSP difference equation to obtain the math description of the DSP system. We want to write the equation for this signal if it is delayed in time by 3T. A direct application of the preceding property gives the following answer, where F z is the z-transform of f n. Writing the z-transform of an equation with shifted functions Problem: Write the z-transform of the following equation of a DSP system. Digital Signal Processing Solution: Taking the z-transform of all the signals in the equation, we get the following equation. Remember it is composed of current and delayed inputs and delayed outputs.
This kind of equation is called a difference equation, and it is the discrete-time equivalent to the analog differential equation. In Chapter 5 we will see that this property of the z-transform will allow us to write the difference equation as an algebraic equation and then solve it, to get an algebraic equation of a digital filter by getting the ratio of the z-transform of the output over the z-transform of the input, which is the transfer function T z.
Earlier we said we would develop this shifting property in another way, which is actually more intuitive. All we need to do is to look at a few simple sampled signals and compare their z-transforms with and without delays in the sampled-time domain. The z-transforms of all these signals are given in the following equations, using the definition given in Equation 4.
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Summary In this chapter, we have introduced the z-transform of a sampled or discrete-time signal. The definition was given, and it was seen to be an algebraic equation. This definition was used to find the z-transform of an impulse signal and a sampled unit step signal. These two z-transforms will be used later to determine two digital filter approximations to analog filters.
One other z-transform of a signal was determined, the z-transform of an exponential signal. This z-transform will be used later to determine the stability of digital filters.
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We also determined one of the most important properties of z-transforms, the shifting property. This property will allow us to turn the difference equation of a digital filter into an algebraic equation and then determine the mathematical description of a digital filter, called its transfer function. Also, if we have determined the mathematical description, or transfer function, of a digital filter, we can use the shifting property to write the difference equation of the filter that is actually coded. This process will be needed in some of the following chapters.
Determine the z-transform of the following sampled signal. Digital Signal Processing 3.
Determine the z-transform for the following sampled signal. Determine the equation of the sampled pulse x n described here, using the sampled unit step function. Determine the equation of the following sampled pulse signal. Given the only nonzero sample values of the following signal, determine the z-transform of the signal. Determine the equation of the following sampled pulse x n , using the sampled unit step function.
Given the only nonzero values of the following signal, determine the z-transform of the signal. Determine the z-transform of the following signal if it were sampled at T - 0. Introduction In this chapter we apply the z-transform to the general DSP equation given in Chapter 1. First the equation is modified to a more useful form, then the z-transform is taken of the discrete time signals on both sides of the equation. The general DSP equation in either form is an input-output difference equation and cannot be solved for the output given the input except by iteration.
By taking the z-transform of the equation, an algebraic equation results that can be solved explicitly for the output given the input.
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The terms multiplying the input to get the output must be the equation of the filter called the transfer function, which is needed to design and analyze DSP systems and digital filters. It must be remembered that it is the general DSP equation that is coded, but it is the mathematical description of the DSP system that is used to analyze and design DSP systems. This equation assumes that a signal sample with no subscript refers to the input or output sampled value at the current time, and a negative integer subscript refers to the corresponding number of sample periods previous Digital Signal Processing to the current time.
This is a convenient and easy way to write the equation for Chapter 1, but here a better and more rigorous equation must be written by making a few changes to Equation 5. For mathematical convenience the sampled signal subscripts will be used as arguments, with the current sample being called the nth sample, so y and x go to y n and x n respectively.
Mathematically this is no change at all, since a subscripted variable is mathematically just a function of an integer subscript. Also x —1 will go to x n — 1 , and other signals will follow this pattern. The sample values of the signals may start at an arbitrary initial sample number 0, and the current sample values refer to the nth sample after this. This will facilitate going back and forth between the analog time t and the discrete time nT to which the nth sample corresponds. Finally, for the coefficients it is standard to drop the negative sign on the subscripts.
Thus the a coefficient goes to a 0, and the b —1 coefficient goes to b 1. The modified DSP equation is shown in Equation 5. Also note that Equations 5.
Example 5. The modification of a DSP equation to the more standard form Problem: Convert the following equation into the modified or standard form. But now if the arguments, which represent sample numbers, are multiplied by the sample period, we have the product representing the actual sample times. Also note that in the preceding equation 3 is b 1, —0. In order to show explicitly how the subscripts of the coefficients are handled to get the modified and more standard form of the general DSP equation, we leave them as arbitrary coefficients in Example 5.