Frequency Domain Analysis of Linear Time-Invariant (LTI) Systems

Input-Output Relations for Linear Time-Invariant Systems via Fourier Analysis

As we discussed in the relevant section, a very important property of convolution systems is that if the input is a harmonic function, so is the output.

Continuous-Time Case. Let $u \in L_\infty(\mathbb{R}; \mathbb{C})$ given with

$$u(t) = e^{i2\pi f t},$$

be the input to a linear time-invariant system

$$y(t) = \int_{\tau=-\infty}^{\infty} h(t - \tau) u(\tau)\, d\tau = \int_{\tau=-\infty}^{\infty} h(\tau) u(t - \tau)\, d\tau$$

Then, the output satisfies

$$y(t) = \left(\int_{-\infty}^{\infty} h(s) e^{-i2\pi f s}\, ds\right) e^{i2\pi f t}$$

The integral

$$\hat{h}(f) := \left(\int h(t) e^{-i2\pi f t}\, dt\right),$$

is $\mathcal{F}_{CC}(h)$ evaluated at $f$. We call this value, the frequency response of the system at frequency $f$, whenever it exists.

Discrete-Time Case. A similar discussion applies for a discrete-time system: Let $h \in l_1(\mathbb{Z}; \mathbb{C})$. If $u(n) = e^{i2\pi f n}$ is the input to a linear time-invariant system given with

$$y(n) = \sum_{m=-\infty}^{\infty} h(n - m) u(m) = \sum_{m=-\infty}^{\infty} h(m) u(n - m).$$

then

$$y(n) = \left(\sum_{m=-\infty}^{\infty} h(m) e^{-i2\pi f m}\right) e^{i2\pi f n}.$$

We recognize that

$$\hat{h}(f) := \sum_{m=-\infty}^{\infty} h(m) e^{-i2\pi f m} = \left(\mathcal{F}_{DC}(h)\right)(f),$$

and call $\hat{h}$ the frequency response function.

Convolution systems are used as filters through the characteristics of the frequency response.

Some Properties.

Recall the following properties of $\mathcal{F}_{CC}$.

(i) Convolution Property. Let $u, v \in \mathcal{S}$. We have that

$$\left(\mathcal{F}_{CC}(u * v)\right)(f) = \hat{u}(f)\hat{v}(f)$$

(ii) Differentiation Property. If $v = \frac{d}{dt}u$, then $\hat{v}(f) = i2\pi f\, \hat{u}(f)$.

(iii) Time-Shift Property for $\mathcal{F}_{DC}$. Let $v = \sigma^\theta(u)$ for some $\theta \in \mathbb{Z}$, that is $v(n) = u(n + \theta)$. Then,

$$\hat{v}(f) = \sum_{n \in \mathbb{Z}} u(n + \theta) e^{-i2\pi f n} = e^{i2\pi \theta f}\, \hat{u}(f)$$

The above will be very useful properties for studying LTI systems. We can also obtain converse differentiation properties, which will be considered in further detail in the relevant section while studying the Z and the Laplace transformations. Nonetheless, we will present two such properties in the following (see Section A.2 for a justification on changing the order of differentiations and summations/integrations):

(iv) Differentiation in Frequency for $\mathcal{F}_{DC}$. Let

$$\mathcal{F}_{DC}(x)(f) = \hat{x}(f) = \sum_n x(n) e^{-i2\pi f n}$$

Then, through changing the order of limit and summation:

$$\frac{d\hat{x}(f)}{df} = \sum_n \frac{dx(n) e^{-i2\pi f n}}{df} = \sum_n (-2i\pi n\, x(n)) e^{-i2\pi f n}$$

This leads to the conclusion that with $v(n) = -n\, x(n)$, with $v$ absolutely summable,

$$\mathcal{F}_{DC}(v)(f) = \frac{1}{i2\pi} \frac{d\hat{x}(f)}{df}$$

(v) Differentiation in Frequency for $\mathcal{F}_{CC}$. Likewise, for the continuous-time case with $x \in \mathcal{S}$

$$\mathcal{F}_{CC}(x)(f) = \hat{x}(f) = \int_t x(t) e^{-i2\pi f t}\, dt$$

Via the analysis in Section A.2, through changing the order of limit and integration,

$$\frac{d\hat{x}(f)}{df} = \int \frac{dx(t) e^{-i2\pi f t}}{df}\, dt = \int (-2i\pi t\, x(t)) e^{-i2\pi t}\, dt$$

This leads to the conclusion that with $v(t) = -t\, x(t)$, with $v(t)$ (absolutely) integrable,

$$\mathcal{F}_{CC}(v)(f) = \frac{1}{i2\pi} \frac{d\hat{x}(f)}{df}$$

Useful Transform Pairs. In the context of LTI systems, we will occasionally build on the following properties:

• If $u(t) = 1_{\{t \geq 0\}} e^{at}$, with $a < 0$, then $\hat{u}(f) = \frac{1}{-a + i2\pi f}$.
• Likewise for $\mathcal{F}_{DC}$, for $|a| < 1$, if $u(n) = a^{n-1} 1_{\{n \geq 1\}}$, then $\hat{u}(f) = e^{-i2\pi f} \frac{1}{1 - a e^{-i2\pi f}}$.

The properties above are crucial, and typically sufficient, for studying a large class of linear time invariant systems described by differential and difference equations (convolution systems).

Transfer Functions and their Computation for Convolution Systems via Fourier Transforms

In applications for control, communications, and signal processing, one may design systems or filters using the properties of the frequency response functions.

Continuous-Time Systems

Consider the following continuous-time system with input $u$ and output $y$:

$$\sum_{k=0}^{N} a_k \frac{d^k}{dt^k} y(t) = \sum_{m=0}^{M} b_m \frac{d^m}{dt^m} u(t)$$

Taking the $\mathcal{F}_{CC}$ of both sides, we obtain

$$\left(\sum_{k=0}^{N} a_k (i2\pi f)^k\right) \hat{y}(f) = \left(\sum_{m=0}^{M} b_m (i2\pi f)^m\right) \hat{u}(f)$$

This leads to:

$$\hat{h}(f) = \frac{\sum_{m=0}^{M} b_m (i2\pi f)^m}{\sum_{k=0}^{N} a_k (i2\pi f)^k}$$

Discrete-Time Systems

Likewise, for discrete-time systems:

$$\sum_{k=0}^{N} a_k\, y(n-k) = \sum_{m=0}^{M} b_m\, u(n-m)$$

Taking the $\mathcal{F}_{DC}$ of both sides (assuming the $\mathcal{F}_{DC}$ exist), we obtain

$$\hat{h}(f) = \frac{\hat{y}(f)}{\hat{u}(f)} = \frac{\sum_{m=0}^{M} b_m e^{-i2\pi m f}}{\sum_{k=0}^{N} a_k e^{-i2\pi k f}}$$

Computing the Inverse Transform via Partial Fractions

How to compute the inverse transform? One can, using $\hat{h}$, compute $h(t)$ or $h(n)$, if one is able to compute the inverse transform. A useful method is the partial fraction expansion method. More general techniques will be discussed in the following chapter. Let

$$R(\lambda) = \frac{P(\lambda)}{Q(\lambda)} = \frac{p_0 + p_1 \lambda + \cdots + p_M \lambda^M}{q_0 + q_1 \lambda + \cdots + q_N \lambda^N}, \quad \lambda \in \mathbb{C}$$

If $M < N$, we call this fraction strictly proper. If $M \leq N$, the fraction is called proper and if $M > N$, it is called improper.

If $M > N$, we can write

$$R(\lambda) = T(\lambda) + \tilde{R}(\lambda),$$

where $T$ has degree $M - N$ and $\tilde{R}(\lambda)$ is strictly proper. We can in particular write:

$$\tilde{R}(\lambda) = \sum_{i=1}^{K} \left(\sum_{k=1}^{m_i} \frac{A_{ik}}{(\lambda - \lambda_i)^k}\right)$$

where $\lambda_i$ are the roots of $Q$ and $m_i$ is the multiplicity of $\lambda_i$.

This is important because we can use the expansion and the properties of $\mathcal{F}_{CC}$ and $\mathcal{F}_{DC}$ presented earlier in the chapter to compute the inverse transforms. Such approaches will be studied in further detail in the following chapter. In the following, we present two examples.

Exercises