Introduction to Fourier representation of signals and Complex sinusoids and frequency response of LTI systems
We will start by representing a signal as a weighted superposition of complex sinusoids. If such a signal is applied to an LTI system, the system’s output is a weighted superposition of the system response to each complex sinusoid.
Representing signals as superpositions of complex sinusoids not only leads to a useful expression for the system output but also provides an insightful characterization of signals and systems. This general notion of describing complicated signals as a function of frequency is commonly encountered in music.
For example, the musical score for an orchestra contains parts for instruments having different frequency ranges, such as a string bass, which produces very low-frequency sound, and a piccolo, which produces very high-frequency sound.
The sound we hear when listening to an orchestra is a superposition of sounds generated by different instruments. In a similar way, the score for a choir contains bass, tenor, alto, and soprano parts, each of which contributes to a different frequency range in the overall sound.
The study of signals and systems using sinusoidal representations is termed Fourier analysis. It has been named after Joseph Fourier (1768-1830) for his development of the theory. Fourier methods have widespread application beyond signals and systems and are used in every branch of engineering and science.
There are four distinct Fourier representations, each applicable to a different class of signals, determined by the periodicity of the signal and whether the signal is discrete or continuous in time.
Complex sinusoids and frequency response of LTI systems
The response of the LTI system to a sinusoidal input leads to a characterization of the system behavior that is termed the frequency response of the system. This characterization is obtained in terms of the impulse response by using convolution and a complex sinusoidal input signal. Consider the output of a discrete-time LTI system with impulse response h[n] and unit amplitude complex sinusoidal input x[n]= e^iΩn. This output is given by:
y[n] = Σh[k]x[n-k] = Σh[k] e^iΩ(n-k).
We factor the e^iΩn from the sum to obtain:
y[n] = e^iΩnΣh[k]x[n-k] = H(e^iΩ)e^iΩn
Where we have defined:
H(e^iΩ)= Σh[k] e^-iΩk.
Therefore, the output of the system is a complex sinusoid of the same frequency as the input multiplied by the complex number H(e^iΩ). The complex scaling factor H(e^iΩ) is not a function of time n, but is only a function of frequency Ω and is termed the frequency response of the discrete-time system.
Similar results are obtained for continuous-time LTI systems. Let the impulse response of such system be h(t) and the input be x(t)= e^iwt. Then the convolution integral of the output is:
y(t) = ∫ h(T)e^iw(t-T) dT. = H(jw) e^iwt.
The output of the system is thus a complex sinusoid of the same frequency as the input, multiplied by the complex number H(jw) is a function of only the frequency w and not the time t and is termed the frequency response of the continuous-time system.
An intuitive interpretation of the sinusoidal steady-state response is obtained by writing the complex-valued frequency response H(jw) in polar form.
As we know that if c= a+ jb is a complex number, then we may write c in polar form as c= |c| e^arg(c) , where c = sqrt(a^2 + b^2) and arg{c} = arctan(b/a). Therefore, we have H(jw) = |H(jw) e^iarg(H(jw))|, where |H(jw)| is now termed as magnitude response and arg{H(jw)} is termed the phase response of the system.
Substituting this polar form in the mentioned equation, we can express the output as:
y(t) = |H(jw)| e^j(wt + arg{H(jw)}).
The system thus modifies the amplitude of the input by |H(jw)| and the phase by arg{H(jw)}.
We also say that the complex sinusoid e^iwt is an eigenfunction of the LTI system H associated with the eigenvalue H(jw), because it satisfies the eigenvalue problem given by:
H{ψ(t)} = λψ(t).
The effect of the system on an eigenfunction input signal is scalar multiplication, that is, the output is given by the product of the input and a complex number. This eigen representation is analogous to the matrix eigenproblem.
If ek is an eigen vector of a matrix A with eigen value λk, then
Aek = λkek.
In other words, pre-multiplying ek by the matrix A is equivalent to multiplying ek by the scalar λk.
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