Fourier series representation of Periodic Signals

1. Response of Linear Time-unchanged system to the complex index Signal

When studying the \ (LTI \) (linear and time-invariant System) system, it is advantageous to represent a linear combination of signals as basic signals, however, these basic signals should have the following two properties:

• These basic signals can form a wide range of useful signals;
• \ (LTI \) the system's response to each basic signal should be very simple, so that the system has a convenient representation of the response to any input signal.

Many important values of Fourier analysis come from this point, that is, the continuous and discrete-time complex exponent signal sets have the above two properties, namely, the continuous \ (E ^ {st }\) and discrete time \ (Z ^ n \), where \ (S \) and \ (Z \) are both plural.

When studying the \ (LTI \) system, the importance of the exponent signal lies in the fact thatThe response of an \ (LTI \) system to the complex index signal is also a complex index signal. The difference is only the amplitude change.That is to say:

\ [Continuous Time: E ^ {st} \ to H (s) e ^ {st} \]
\ [Discrete Time: Z ^ {n} \ to H (z) Z ^ {n} \]
Here \ (H (s) \) or \ (H (z) \) is a complex amplitude factor, which is generally a complex variable \ (S \) or \ (Z \). A signal. If the system returns only one constant multiplied by the input, it is called the system'sFeature FunctionsThe amplitude factor is called the system'sFeature value.

Consider a continuous time \ (LTI \) system with an impulse response of \ (h (T) \), for any input \ (X (t )\), the output can be determined by convolution credits. If the \ (X (t) = E ^ {st} \),

\ [\ Tag 1 y (t) = \ int _ {-\ infty} ^ {+ \ infty} H (\ Tau) x (t-\ Tau) d \ Tau = \ int _ {-\ infty} ^ {+ \ infty} H (\ Tau) e ^ {S (t-\ Tau )} d \ Tau = e ^ {st} \ int _ {-\ infty} ^ {+ \ infty} H (\ Tau) e ^ {-S \ Tau} d \ Tau \]

Assuming that the credits on the right side of formula (1) converge, the system's response to \ (X (t) \) is
\ [\ Tag 2 Y (t) = H (s) e ^ {st} \]
In formula \ (H (s) \) is a complex constant whose value is determined by \ (S \) and its relationship with the impulse response of the system unit is
\ [\ Tag 3 h (s) =\int _ {-\ infty} ^ {+ \ infty} H (\ Tau) e ^ {-S \ Tau} d \ Tau \]

It can be proved completely in parallel that the complex index sequence is also a feature function of the discrete time \ (LTI \) system. This means that the unit pulse response is \ (H [N] \) \ (LTI \) system, and its input sequence is
\ [\ Tag 4 x [N] = Z ^ {n} \]
In formula \ (Z \) is a complex number, and the output of the system is determined by convolution and
\ [\ Tag 5 y [N] = \ sum _ {k =-\ infty} ^ {+ \ infty} H [k] X [n-k] = \ sum _ {k =-\ infty} ^ {+ \ infty} H [k] Z ^ {n-k} = Z ^ n \ sum _ {k =-\ infty} ^ {+ \ infty} H [k] Z ^ {-k} \]
Assume that the sum on the right side of formula (5) converges, so the system's response to \ (X [N] \) is
\ [\ Tag 6 y [N] = H (z) Z ^ {n} \]
In formula \ (H (z) \) is a complex constant
\ [\ Tag 7 H [Z] = \ sum _ {k =-\ infty} ^ {+ \ infty} H [k] Z ^ {-k} \]

For more general cases, if the input of a continuous time \ (LTI \) system is expressed as a linear combination of the complex index, that is
\ [\ Tag 8 x (t) = \ sum_k A_k e ^ {s_kt} \]
The output must be
\ [\ Tag 9 y (t) = \ sum_k a_kh (S_k) e ^ {s_kt} \]

For discretization, It is similar. If the input of a discrete time \ (LTI \) system is expressed as a linear combination of the complex index, that is
\ [\ Tag {10} X [N] = \ sum_k A_k z_k ^ n \]
The output must be
\ [\ Tag {11} y [N] = \ sum_k a_kh (z_k) z_k ^ n \]

2. Linear Combination of Fourier series of continuous time-cycle signals, which represents 2.1. complex exponential signals with harmonic relationships

Periodic exponential response signal
\ [\ Tag {12} x (t) = E ^ {J \ omega_0 t} \]
The fundamental frequency is \ (\ omega_0 \), and the fundamental frequency \ (t = 2 \ PI/\ omega_0 \). RelatedHarmonic relationshipThe complex index Signal Set of is
\ [\ Tag {13} \ phi_k (t) = E ^ {j k \ omega_0 t} = e ^ {j k (2 \ PI/T) t }, k = 0, \ PM1, \ PM2, \ cdot \]

Each of these signals has a fundamental frequency, which is a multiple of \ (\ omega_0. Therefore, each signal is periodic for cycle \ (T. Therefore, a signal is formed by a linear combination of complex exponential signals that generate harmonic relationships.
\ [\ Tag {14} x (t) = \ sum _ {k =-\ infty} ^ {\ infty} a_ke ^ {j k \ omega_0 t} = \ sum _ {k =-\ infty} ^ {\ infty} a_ke ^ {j k (2 \ PI/T) t} \]
Cycle \ (T \) is also a cycle. In formula (14), \ (k = 0 \) is a constant, \ (k = + 1 \) and \ (k =-1 \) both of these items have a fundamental frequency equal to \ (\ omega_0 \).Fundamental componentOrHarmonic Component. \ (K = + 2 \) and \ (k =-2 \) are also periodic, and their frequency is twice the fundamental frequency.Second Harmonic Component. In general, the \ (k = + n \) and \ (k =-n \) components are called the \ (n \) subharmonic components.

A periodic signal is expressed in the form of formula (14), which is calledFourier Series.

2.2. Determination of Fourier series representation of Continuous Time Periods

Assume that a given periodic signal can be expressed in the form of formula (14), which requires a way to determine these coefficients \ (A_k \), convert formula (14) multiply each side by \ (E ^ {-JN \ omega_0t} \).
\ [\ Tag {15} x (t) e ^ {-JN \ omega_0t} = \ sum _ {k =-\ infty} ^ {\ infty} a_ke ^ {j k \ omega_0 t} e ^ {-JN \ omega_0t} \]
Convert the above formula from 0 to \ (t = 2 \ PI/\ omega_0 \) to \ (T \) points,
\ [\ Tag {16} \ int _ 0 ^ TX (t) e ^ {-JN \ omega_0t} dt = \ int _ 0 ^ t \ sum _ {k =-\ infty} ^ {\ infty} a_ke ^ {j k \ omega_0 t} e ^ {-JN \ omega_0t} DT \]

Here \ (T \) is the fundamental cycle of \ (X (t) \), and the above is the point in this cycle. After the points and the sum order on the right side of the above formula are exchanged
\ [\ Tag {17} \ int _ 0 ^ TX (t) e ^ {-JN \ omega_0t} dt = \ sum _ {k =-\ infty} ^ {\ infty} A_k \ int _ 0 ^ te ^ {J (k-N) \ omega_0 t} DT \]
It is easy to use the integral points in the right brace of formula (17 ).

\ [\ Tag {18} \ int _ 0 ^ te ^ {J (k-N) \ omega_0 t} dt = \ int _ 0 ^ t cos (k-N) \ omega_0 TdT + J \ int _ 0 ^ t sin (k-N) \ omega_0 TDT \]

For \ (k \ not = n \), \ (COS (k-N) \ omega_0 t \) and \ (sin (k-N) \ omega_0 t \) all are periodic functions, and their fundamental frequency is \ (T/| K-N | )\). The current credits are performed in the \ (T \) interval, and \ (T \) must be their fundamental frequency \ (T/| K-N |).Points can be regarded as the area included by the product function in the integral range. Therefore, for the two points on the Right of formula (18), \ (k \ not = n, the value is 0.While for \ (k = n \), the product function on the left side of the formula is 1, so its integral value is \ (T \). Based on the above results
\ [\ Tag {19} \ int _ 0 ^ te ^ {J (k-N) \ omega_0 t} dt = \ begin {cases} t, & \ Text K = n \ 0, & \ Text K \ not = n \ end {cases} \]
In this way, the right side of formula (17) is changed to \ (ta_n \). Therefore

\ [\ Tag {20} a_n = \ frac {1} {t} \ int _ 0 ^ TX (t) e ^ {-j n \ omega_0 t} DT \]

In addition, when formula (18) is used, we only use the credits in a \ (T \) interval, and the \ (T \) it is an integer multiple of the \ (COS (k-N) \ omega_0 t \) and \ (sin (k-N) \ omega_0 t \) cycles. Therefore,If the points are made at any \ (T \) interval, the result should be the same.That is to say, if \ (\ int _ t \) represents the points within any \ (T \) interval, there should be

\ [\ Tag {21} \ int _ te ^ {J (k-N) \ omega_0 t} dt = \ begin {cases} t, & \ Text K = n \ 0, & \ Text K \ not = n \ end {cases} \]
Therefore
\ [\ Tag {22} a_n = \ frac {1} {t} \ int _ TX (t) e ^ {-j n \ omega_0 t} DT \]

The above process can be concluded as follows: If \ (X (t) \) can represent a linear combination of the complex index signals that constitute the harmonic relationship, then the coefficients in the Fourier series are expressed by formula (22) as determined, this relationship is defined as the Fourier count of a periodic continuous signal.
\ [\ Boxed {x (t) = \ sum _ {k =-\ infty} ^ {\ infty} a_ke ^ {j k \ omega_0 t} = \ sum _ {k =-\ infty} ^ {\ infty} a_ke ^ {j k (2 \ PI/T) t} \ A_k = \ frac {1} {t} \ int _ TX (t) e ^ {-j k \ omega_0 t} dt = \ frac {1} {t} \ int _ TX (t) e ^ {-j k (2 \ PI/T) t} DT} \]
The first statement is calledComprehensive FormulaThe second formula is calledAnalysis formula. The coefficient \ ({A_k} \) is often called \ (X (t) \)Fourier series coefficient or Spectrum Coefficient.

• Example 1
  
  
• Example 2
  
  
  

2.3. Convergence of Fourier Series

For any cyclic signal, we can always use formula (22) to obtain a set of Fourier coefficients. However, in some cases, the integral of formula (22) may not converge, that is, some obtained coefficients may be infinite. Furthermore, even if all the obtained coefficients are finite values, the infinite series obtained when these coefficients are substituted into formula (14) may not converge to the original signal.

Dinheri Condition:

1. In any period, \ (X (t) \) must be absolutely product-ready, That is
   \ [\ Tag {23} \ int_t | x (t) | dt <\ infty \]
   This condition ensures that each coefficient \ (A_k \) is finite because
   \ [\ Tag {24} | A_k | \ leqslant \ frac {1} {t} \ int_t | x (t) e ^ {JK \ omega_0t} | dt = \ frac {1} {t} \ int_t | x (t) | DT \]
   The following is an example of a periodic signal that does not meet the first piece of di liheri:
   \ [\ Tag {25} x (t) = \ frac {1} {t}, 0 <t \ leqslant1 \]
   

2. In any finite range, \ (X (t) \) has limited fluctuations, that is, in any single cycle, \ (X (t )\) the maximum and minimum values of are limited..

A function that satisfies condition 1 but does not meet condition 2 is
\ [\ Tag {26} x (t) = sin (\ frac {2 \ PI} {t}), 0 <t \ leqslant1 \]

1. In any finite interval of \ (X (t) \), there are only finite discontinuous points. In addition, the functions are finite values on these discontinuous points..

An example of not meeting Condition 3 is as follows. The signal cycle is \ (t = 8 \), which is composed of the following: the height and width of the last step are half of the previous step.

2.4. Properties of Fourier Series

3. Fourier series of discrete time period signals represents a linear combination of complex exponent signals in a harmonic relationship of 3.1.

Periodic exponential response signal
\ [\ Tag {27} X [N] = e ^ {J (2 \ PI/n) n} \]
The fundamental frequency is \ (\ omega_0 = 2 \ PI/n \), and the fundamental frequency is \ (n \). RelatedHarmonic relationshipThe complex index Signal Set of is
\ [\ Tag {28} \ phi_k [N] = e ^ {j k \ omega_0 n} = e ^ {j k (2 \ PI/n) n }, k = 0, \ PM1, \ PM2, \ cdot \]

Each of these signals has a fundamental frequency, which is a multiple of \ (2 \ PI/n. Formula (28)Only \ (n \) signals in the given signal set are different, because the frequency difference of the discrete time complex exponent signal \ (2 \ PI/n \) the integer multiples of are the same.. Therefore
\ [\ Tag {29} \ phi_k [N] = \ Phi _ {k + rn} [N] \]
That is to say, when \ (k \) changes to an integer multiple of \ (n \), it gets a completely identical sequence. Now we want to use the linear combination of the sequence \ (\ phi_k [N] \) to represent a more general periodic sequence. Such a linear combination has the following form:
\ [\ Tag {30} X [N] = \ sum _ {k} A_k \ phi_k [N] = \ sum _ {k} a_ke ^ {j k \ omega_0 n} = \ sum _ {k} a_ke ^ {j k (2 \ PI/N) n} \]
Because the sequence \ (\ phi_k [N] \) only has different \ (n \) successive values in \ (k \), the formula (30) the summation of only the \ (n \) items must be included. To point this out, the summation limit is expressed as \ (k = <n> \), that is
\ [\ Tag {31} X [N] = \ sum _ {k = <n >}a_k \ phi_k [N] = \ sum _ {k = <n>} a_ke ^ {j k \ omega_0 n }=\ sum _ {k = <n >}a_ke ^ {j k (2 \ PI/N) n} \]
For example, \ (k \) can take \ (k = 0, 1, 2, \ cdot, N-1 \), or \ (k = 3, 4, \ cdot, N + 2) the sum on the right is the same. Formula (31) is calledDiscrete Time Fourier SeriesWhile the coefficient is calledFourier series Coefficient.

3.2. Determination of discrete time period Fourier series Representation

Discrete Time Fourier series PairsFor
\ [\ Boxed {x [N] = \ sum _ {k = <n >}a_ke ^ {j k \ omega_0 n }=\ sum _ {k = <n>} a_ke ^ {j k (2 \ PI/N) n} \ A_k = \ frac {1} {n} \ sum _ {k = <n>} X [N] e ^ {-j k \ omega_0 n} = \ frac {1} {t} \ sum _ {k = <n>} X [N] e ^ {-j k (2 \ PI/N) N }}\]
The first formula is calledComprehensive FormulaThe second formula is calledAnalysis formula. The coefficient \ ({A_k} \) is often called \ (X [N] \)Spectrum Coefficient.

Return to formula (31) and we can see that if \ (k \) is obtained from 0 to \ (N-1 \),
\ [\ Tag {32} X [N] = a_0 \ phi_0 [N] + a_1 \ phi_1 [N] + \ cdot + A _ {N-1} \ Phi _ {N-1} [N] \]
Similarly, if \ (k \) is obtained from 1 to \ (n \),
\ [\ Tag {33} X [N] = A_1 \ phi_1 [N] + A_2 \ phi_2 [N] + \ cdot + A _ {n} \ Phi _ {n} [N] \]
Because $ \ phi_0 [N] = \ phi_n [N] $, the formula (32) and formula (33) are compared, \ (a_0 = A _ {n }\). Similarly, if \ (k \) gets any group of \ (n \) connected integers, there must be
\ [\ Tag {34} A_k = A _ {k + n} \]
That is to say,If the \ (k \) value we are considering is redundant \ (n \), then the \ (A_k \) value must be in \ (n \) as the cycle, cyclical repetition.

• Example 1
  

3.3. Properties of Discrete Time Fourier Series

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Fourier series representation of Periodic Signals