Summary of LTI Systems: Frequency Domain Analysis
LTI Systems: Frequency Domain Analysis – A Student Guide
Introduction
This material explains how sinusoidal signals (continuous- and discrete-time) are represented and how they behave when used to probe Linear Time-Invariant (LTI) systems. We focus on signal definitions, analytic (complex) sinusoids, phasor interpretation, and how an LTI system transforms a sinusoid in the frequency domain. The presentation breaks concepts into small steps and includes examples and applications.
Definition: A sinusoidal signal is a time-varying function of the form $A\cos\left(\omega t + \varphi\right)$ where $A$ is amplitude, $\omega$ is angular frequency, and $\varphi$ is initial phase.
1. Continuous-Time Sinusoidal Signals
1.1 Real-valued sinusoid
A real-valued continuous-time sinusoid is
$$s(t) = A\cos\left(\omega t + \varphi\right)$$
- Parameters: amplitude $A$, angular frequency $\omega$, frequency $f = \frac{\omega}{2\pi}$, period $T = \frac{1}{f}$, initial phase $\varphi$.
- It is a fundamental signal used to analyze LTI systems because of the system’s invariance properties.
Definition: Period $T$ satisfies $s(t+T) = s(t)$ for all $t$; frequency $f = \frac{1}{T}$.
1.2 Complex (analytic) sinusoid and phasors
A more convenient representation for analysis is the complex sinusoid (analytic signal):
$$\hat{s}(t) = A e^{j(\omega t + \varphi)}$$
- Real signal relates by $\operatorname{Re}{\hat{s}(t)} = A\cos\left(\omega t + \varphi\right)$.
- The complex sinusoid can be viewed in two ways:
- Cartesian: separate real and imaginary parts $A\cos(\omega t + \varphi)$ and $A\sin(\omega t + \varphi)$.
- Polar: a rotating phasor of magnitude $A$ and angle $\omega t + \varphi$.
2. Discrete-Time Sinusoidal Signals
Definition: A discrete-time sinusoid is obtained by sampling a continuous sinusoid or defined directly on integer indices $n$.
2.1 Real-valued discrete sinusoid
$$s[n] = A\cos\left(\Omega n + \varphi\right), \quad n \in \mathbb{Z}$$
- Here $\Omega$ is the normalized angular frequency in radians per sample.
- Sampling a continuous sinusoid with sampling period $T_s$ gives $\Omega = \omega T_s$.
2.2 Analytic discrete sinusoid
$$\hat{s}[n] = A e^{j(\Omega n + \varphi)}$$
- Again, real part recovers the real-valued sinusoid.
- Visualizations: Cartesian plots of real and imaginary parts or a phasor that rotates in steps at each sample index $n$.
3. LTI Systems and Sinusoids (Frequency-Domain View)
Definition: An LTI system is one that is both linear and time-invariant; it is fully characterized by its impulse response $g(t)$ in continuous time or $g[n]$ in discrete time.
3.1 Action of an LTI system on an analytic sinusoid
When an analytic sinusoid is input to an LTI system, the output is the same sinusoid multiplied by a complex constant that depends on frequency.
Continuous-time input:
$$\hat{x}(t) = A e^{j(\omega t + \varphi)}$$
If the system has frequency response $H(j\omega)$, the output is
$$\hat{y}(t) = H(j\omega),A e^{j(\omega t + \varphi)}$$
Discrete-time input:
$$\hat{x}[n] = A e^{j(\Omega n + \varphi)}$$
If the system has transfer function $G(z)$ and we evaluate it on the unit circle $z = e^{j\Omega}$, write $G(e^{j\Omega})$, then
$$\hat{y}[n] = G\left(e^{j\Omega}\right),A e^{j(\Omega n + \varphi)}$$
- The magnitude $\left|G(e^{j\Omega})\right|$ scales the amplitude.
- The argument $\arg\left(G(e^{j\Omega})\right)$ adds to the input phase, producing a phase shift.
Definition: Frequency response is the system’s transfer function evaluated on the imaginary axis (continuous) or on the unit circle (discrete).
3.2 Interp
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Sinusoids and LTI Analysis
Klíčové pojmy: A continuous sinusoid: $s(t)=A\cos\left(\omega t+\varphi\right)$ with $f=\frac{\omega}{2\pi}$ and $T=\frac{1}{f}$, Analytic sinusoid: $\hat{s}(t)=Ae^{j(\omega t+\varphi)}$, real part gives physical signal, Discrete sinusoid: $s[n]=A\cos\left(\Omega n+\varphi\right)$, where $\Omega=\omega T_s$ when sampled, LTI systems scale and shift complex exponentials: $\hat{y}=H(j\omega)\hat{x}$ (continuous) or $\hat{y}=G(e^{j\Omega})\hat{x}$ (discrete), Magnitude $|H(j\omega)|$ scales amplitude; $\arg(H(j\omega))$ adds phase shift, Phasor representation: input rotating phasor multiplied by complex constant gives output phasor, To measure frequency response, sweep frequency and record output amplitude and phase, Use small amplitudes to keep system in linear range; sample properly to avoid aliasing, RC low-pass example: $H(j\omega)=\frac{1}{1+j\omega RC}$, magnitude $\frac{1}{\sqrt{1+(\omega RC)^2}}$, Evaluate discrete transfer function on unit circle: $G(e^{j\Omega})$ gives frequency behavior