9-4 Eigen Functions (T)

Old Chinese version

If the output of a system has the same type as its input signal, then the input signal is referred to as the eigen function of the system. In this section, we shall demonstrate that the exponential function (including the sinusoidal function as a special case) is an eigen function of the LTI system.

Suppose that the input signal to our LTI system is an exponential function:

x[n]=e^pn Then the output can be derived based on convolution:

y[n]	=	x[n]*h[n]
	=	Skx[n-k] h[k]
	=	Skep(n-k) h[k]
	=	Skepn-pk h[k]
	=	epnSke-pk h[k]
	=	epnH(ep)

In other words, the output signal y[n] is equal to the original input signal multiplied by a constant H(e^p)（= S_ke^-pk h[k], which is not a function of time）. Hence it can be established that the exponential function is the eigen function of an LTI system.

Based on the above fact, for any given input signal that can be decomposed as a linear combination of exponential functions, we can then apply the superposition principle to find the responses of these exponential functions first, and then linearly combine these responses to obtain the final output. In mathematical notations, we have:

x[n] = ae^pn + be^qn → y[n] = aH(e^p) + bH(e^q)

Since the exponential function and the sinusoidal function are related by the famous Euler identity:

e^jq = cos(q) + j sin(q) Therefore we can express the sinusoidal function as a linear combination of the exponential function: cos(q) = (e^jq + e^-jq)/2 = Re{e^jq}
sin(q) = (e^jq - e^-jq)/(2j) = Im{e^jq} As a result, when the input signal is: x[n] = cos(wn) = Re{e^jwn} The corresponding response is:

y[n]	=	Re{ejwnH(ejw)}
	=	Re{ejwn |H(ejw)| ejq}, q = ∠H(ejw)
	=	Re{|H(ejw)| ej(wn+q)}
	=	|H(ejw)| cos(wn + q)

From the above derivation, we can see that the output is still a sinusoidal function, except that the amplitude is multiplied by |H(e^jw)|, and the phase is shifted by q = ∠H(e^jw).

Some terminologies are explained next.

• The multiplier |H(e^jw)| represents how the system amplifies or compresses the input signal depending on the angular frequency w. Hence |H(e^jw)| is called magnitude frequency response.
• ∠H(e^jw) is called phase frequency response, which is not used often since human's aural perception is not sensitive to phase shift.
• H(e^jw) is called the system's frequency response.

Similarly, when the input signal is x[n] = sin(wn) = Im{e^jwn} The output can be derived as follows:

y[n]	=	Im{ejwnH(ejw)}
	=	Im{ejwn |H(ejw)| ejq}, q = ∠H(ejw)
	=	Im{|H(ejw)| ej(wn+q)}
	=	|H(ejw)| sin(wn + q)

In summary, we have the following important conclusion. For an LTI system, if the input signal is a single sinusoidal function, the output is also a same-frequency sinusoidal function, with the amplitude and phase modified by the system's frequency response H(e^jw).

In general, we can use an complex exponential function to subsume the input of sine and cosine functions:

x[n] = e^jwn The corresponding output is y[n] = e^jwnH(e^jw) Alternatively, H(e^jw) is also referred to as the the discrete-time Fourier transform of h[n], which can be expressed as the following equation: H(e^jw) = S_kh[k]e^-jwk In the real world, we cannot have the complex exponential function as the input to a system. However, the derivation based on complex numbers does hold mathematically and it indeed makes our derivation more concise.

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Audio Signal Processing and Recognition (音訊處理與辨識)