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Causal discrete-time signals that are linear combinations of real or complex exponentials do have rational transfer functions. However, not all causal discrete-time signals are linear combinations of real or complex exponentials.

For example, consider the causal LTI system whose (infinite) impulse response is

$$h (n) = \begin{cases} \frac{1}{1+n} & \text{ if } n \geq 0\\ \,\,\,0 & \text{ if } n < 0\end{cases}$$

Taking the Z-transform, we obtain the following non-rational transfer function

$$H (z) = \sum_{n=0}^{\infty} \frac{z^{-n}}{1+n} = - z \ln \left(\frac{z-1}{z}\right)$$

when $|z| \geq 1$ and $z \neq 1$. When $z=1$, we have the (divergent) harmonic series.

Can this LTI system be implemented? Using finite-precision arithmetic, $h (n)$ will eventually underflow at some very large $n$. Hence, we can truncate the infinite impulse response $h$, which produces an FIR filter that requires an astronomically long cascade of delays.

Of course, the same underflow would happen if we had the causal infinite impulse response $2^{-n}$. However, $2^{-n}$ is a real exponential and can be produced by the 1st order difference equation

$$y (n) - \frac 12 y (n-1) = x (n)$$

which requires only $1$ adder, $1$ multiplier and $1$ delay.

Exponentials, whether real or complex, have low complexity, i.e., they can be generated using few adders, multipliers and delays. Using Fourier transforms, signals of interest can be approximated by linear combinations of complex exponentials.

Causal discrete-time signals that are linear combinations of real or complex exponentials do have rational transfer functions. However, not all causal discrete-time signals are linear combinations of real or complex exponentials.

For example, consider the causal LTI system whose (infinite) impulse response is

$$h (n) = \begin{cases} \frac{1}{1+n} & \text{ if } n \geq 0\\ \,\,\,0 & \text{ if } n < 0\end{cases}$$

Taking the Z-transform, we obtain the following non-rational transfer function

$$H (z) = \sum_{n=0}^{\infty} \frac{z^{-n}}{1+n} = - z \ln \left(\frac{z-1}{z}\right)$$

when $|z| \geq 1$ and $z \neq 1$. When $z=1$, we have the (divergent) harmonic series.

Can this LTI system be implemented? Using finite-precision arithmetic, $h (n)$ will eventually underflow at some very large $n$. Hence, we can truncate the infinite impulse response $h$, which produces an FIR filter that requires an astronomically long cascade of delays.

Of course, the same underflow would happen if we had the causal infinite impulse response $2^{-n}$. However, $2^{-n}$ is a real exponential and can be produced by the 1st order difference equation

$$y (n) - \frac 12 y (n-1) = x (n)$$

which requires only $1$ adder, $1$ multiplier and $1$ delay.

Exponentials — whether real or complex — have low complexity, i.e., they can be generated using few adders, multipliers and delays. Do note that discrete-time exponentials are eigenfunctions of the delay operator.

Causal discrete-time signals that are linear combinations of real or complex exponentials do have rational transfer functions. However, not all causal discrete-time signals are linear combinations of real or complex exponentials.

For example, consider the causal LTI system whose (infinite) impulse response is

$$h (n) = \begin{cases} \frac{1}{1+n} & \text{ if } n \geq 0\\ \,\,\,0 & \text{ if } n < 0\end{cases}$$

Taking the Z-transform, we obtain the following non-rational transfer function

$$H (z) = \sum_{n=0}^{\infty} \frac{z^{-n}}{1+n} = - z \ln \left(\frac{z-1}{z}\right)$$

when $|z| \geq 1$ and $z \neq 1$. When $z=1$, we have the (divergent) harmonic series.

Can this LTI system be implemented? Using finite-precision arithmetic, $h (n)$ will eventually underflow at some very large $n$. Hence, we can truncate the infinite impulse response $h$, which produces an FIR filter that requires an astronomically long cascade of delays.

Of course, the same underflow would happen if we had the causal infinite impulse response $2^{-n}$. However, $2^{-n}$ is a real exponential and can be produced by the 1st order difference equation

$$y (n) - \frac 12 y (n-1) = x (n)$$

which requires only $1$ adder, $1$ multiplier and $1$ delay.

Exponentials, whether real or complex, have low complexity, i.e., they can be generated using few adders, multipliers and delays. Using Fourier transforms, signals of interest can be approximated by linear combinations of real or complex exponentials. If the approximation error is "small", one can then choose to call it "noise" and sweep it under the rug.

Causal discrete-time signals that are linear combinations of real or complex exponentials do have rational transfer functions. However, not all causal discrete-time signals are linear combinations of real or complex exponentials.

For example, consider the causal LTI system whose (infinite) impulse response is

$$h (n) = \begin{cases} \frac{1}{1+n} & \text{ if } n \geq 0\\ \,\,\,0 & \text{ if } n < 0\end{cases}$$

Taking the Z-transform, we obtain the following non-rational transfer function

$$H (z) = \sum_{n=0}^{\infty} \frac{z^{-n}}{1+n} = - z \ln \left(\frac{z-1}{z}\right)$$

when $|z| \geq 1$ and $z \neq 1$. When $z=1$, we have the (divergent) harmonic series.

Can this LTI system be implemented? Using finite-precision arithmetic, $h (n)$ will eventually underflow at some very large $n$. Hence, we can truncate the infinite impulse response $h$, which produces an FIR filter that requires an astronomically long cascade of delays.

Of course, the same underflow would happen if we had the causal infinite impulse response $2^{-n}$. However, $2^{-n}$ is a real exponential and can be produced by the 1st order difference equation

$$y (n) - \frac 12 y (n-1) = x (n)$$

which requires only $1$ adder, $1$ multiplier and $1$ delay.

Exponentials, whether real or complex, have low complexity, i.e., they can be generated using few adders, multipliers and delays. Using Fourier transforms, signals of interest can be approximated by linear combinations of complex exponentials.

Causal discrete-time signals that are linear combinations of real or complex exponentials do have rational transfer functions. However, not all causal discrete-time signals are linear combinations of real or complex exponentials.

For example, consider the causal LTI system whose (infinite) impulse response is

$$h (n) = \begin{cases} \frac{1}{1+n} & \text{ if } n \geq 0\\ \,\,\,0 & \text{ if } n < 0\end{cases}$$

Taking the Z-transform, we obtain the following non-rational transfer function

$$H (z) = \sum_{n=0}^{\infty} \frac{z^{-n}}{1+n} = - z \ln \left(\frac{z-1}{z}\right)$$

Can this LTI system be implemented? Using finite-precision arithmetic, $h (n)$ will eventually underflow at some very large $n$. Hence, we can truncate the infinite impulse response $h$, which produces an FIR filter that requires an astronomically long cascade of delays.

Of course, the same underflow would happen if we had the causal infinite impulse response $2^{-n}$. However, $2^{-n}$ is a real exponential and can be produced by the 1st order difference equation

$$y (n) - \frac 12 y (n-1) = x (n)$$

which requires only $1$ adder, $1$ multiplier and $1$ delay.

Exponentials, whether real or complex, have low complexity, i.e., they can be generated using few adders, multipliers and delays. Using Fourier transforms, signals of interest can be approximated by linear combinations of real or complex exponentials. If the approximation error is "small", one can then choose to call it "noise" and sweep it under the rug.

Causal discrete-time signals that are linear combinations of real or complex exponentials do have rational transfer functions. However, not all causal discrete-time signals are linear combinations of real or complex exponentials.

For example, consider the causal LTI system whose (infinite) impulse response is

$$h (n) = \begin{cases} \frac{1}{1+n} & \text{ if } n \geq 0\\ \,\,\,0 & \text{ if } n < 0\end{cases}$$

Taking the Z-transform, we obtain the following non-rational transfer function

$$H (z) = \sum_{n=0}^{\infty} \frac{z^{-n}}{1+n} = - z \ln \left(\frac{z-1}{z}\right)$$

when $|z| \geq 1$ and $z \neq 1$. When $z=1$, we have the (divergent) harmonic series.

Can this LTI system be implemented? Using finite-precision arithmetic, $h (n)$ will eventually underflow at some very large $n$. Hence, we can truncate the infinite impulse response $h$, which produces an FIR filter that requires an astronomically long cascade of delays.

Of course, the same underflow would happen if we had the causal infinite impulse response $2^{-n}$. However, $2^{-n}$ is a real exponential and can be produced by the 1st order difference equation

$$y (n) - \frac 12 y (n-1) = x (n)$$

which requires only $1$ adder, $1$ multiplier and $1$ delay.

Exponentials, whether real or complex, have low complexity, i.e., they can be generated using few adders, multipliers and delays. Using Fourier transforms, signals of interest can be approximated by linear combinations of real or complex exponentials. If the approximation error is "small", one can then choose to call it "noise" and sweep it under the rug.