附录A 巴塞伐尔定理

1. 能量信号的巴塞伐尔 (Parseval) 定理

设 \(x(t)\) 是一个能量信号， \(x^{*}(t)\) 是 \(x(t)\) 的共轭函数，则有

\[ \int_ {- \infty} ^ {\infty} x ^ {*} (t) h (t + \tau) \mathrm{d} t = \int_ {- \infty} ^ {\infty} x ^ {*} (t) \left[ \int_ {- \infty} ^ {\infty} H (f) \mathrm{e} ^ {\mathrm{j} \omega (t + \tau)} \mathrm{d} f \right] \mathrm{d} t \]

\[ = \int_ {- \infty} ^ {\infty} \left[ \int_ {- \infty} ^ {\infty} x ^ {*} (t) \mathrm{e} ^ {\mathrm{j} \omega t} \mathrm{d} t \right] H (f) \mathrm{e} ^ {\mathrm{j} \omega \tau} \mathrm{d} f = \int_ {- \infty} ^ {\infty} X ^ {*} (f) H (f) \mathrm{e} ^ {\mathrm{j} \omega \tau} \mathrm{d} f \tag {附A-1} \]

式 (附 A-1) 对于任何 \(\tau\) 值都正确，所以可以令 \(\tau=0\) 。这样，式 (附 A-1) 可以化简成

\[ \int_ {- \infty} ^ {\infty} x ^ {*} (t) h (t) \mathrm{d} t = \int_ {- \infty} ^ {\infty} X ^ {*} (f) H (f) \mathrm{d} f \qquad \qquad (\text {附A-2}) \]

若 \(x(t)=h(t)\) ，则式 (附 A-2) 可以改写为

\[ \int_ {- \infty} ^ {\infty} | x (t) | ^ {2} \mathrm{d} t = \int_ {- \infty} ^ {\infty} | X (f) | ^ {2} \mathrm{d} f \qquad \qquad (\text {附A-3}) \]

若 \(x(t)\) 为实函数，则式 (附 A-3) 可以写为

\[ \int_ {- \infty} ^ {\infty} x ^ {2} (t) \mathrm{d} t = \int_ {- \infty} ^ {\infty} | X (f) | ^ {2} \mathrm{d} f \qquad \qquad (\text {附A-4}) \]

式 (附 A-3) 是能量信号的巴塞伐尔定理；式 (附 A-4) 是实能量信号的巴塞伐尔定理。能量信号的巴塞伐尔定理表明，由于一个实信号平方的积分，或一个复信号振幅平方的积分，等于信号的能量，所以信号频谱密度的模的平方 \(\left|X(f)\right|^{2}\) 对 f 的积分也等于信号能量。故称 \(\left|X(f)\right|^{2}\) 为信号的能量谱密度。

2. 周期性功率信号的巴塞伐尔定理

设 \(x(t)\) 是周期性实功率信号，周期等于 \(T_{0}\) , 基频为 \(f_{0}=1/T_{0}\) , 则其傅里叶级数展开

式为

\[ x (t) = \sum_ {n = - \infty} ^ {\infty} C _ {n} \mathrm{e} ^ {\mathrm{j} 2 \pi n f _ {0} t} \tag {附A-5} \]

所以，其平均功率可以写为

\[ \begin{array}{l} \frac {1}{T _ {0}} \int_ {- T _ {0} / 2} ^ {T _ {0} / 2} x ^ {2} (t) \mathrm{d} t = \frac {1}{T _ {0}} \int_ {- T _ {0} / 2} ^ {T _ {0} / 2} x (t) \left[ \sum_ {n = - \infty} ^ {\infty} C _ {n} \mathrm{e} ^ {\mathrm{j} 2 \pi n f _ {0} t} \right] \mathrm{d} t \\ = \sum_ {n = - \infty} ^ {\infty} C _ {n} \frac {1}{T _ {0}} \int_ {- T _ {0} / 2} ^ {T _ {0} / 2} x (t) \mathrm{e} ^ {\mathrm{j} 2 \pi n f _ {0} t} \mathrm{d} t = \sum_ {n = - \infty} ^ {\infty} C _ {n} \cdot C _ {n} ^ {*} = \sum_ {n = - \infty} ^ {\infty} | C _ {n} | ^ {2} \\ \end{array} \]

(附 A - 6)

式 (附 A-6) 就是周期性功率信号的巴塞伐尔定理。它表示周期性功率信号的平均功率等于其频谱的模的平方和。

附录 A 巴塞伐尔定理

\[ \operatorname{erf} (x) = \frac {2}{\sqrt {\pi}} \int_ {0} ^ {x} \mathrm{e} ^ {- z ^ {2}} \mathrm{d} z \]

x	0	1	2	3	4	5	6	7	8	9
1.00	0.84270	84312	84353	84394	84435	84477	84518	84559	84600	84640
1.01	0.84681	84722	84762	84803	84843	84883	84924	84964	85004	85044
1.02	0.85084	85124	85163	85203	85243	85282	85322	85361	85400	85439
1.03	0.85478	85517	85556	85595	85634	85673	85711	85750	85788	85827
1.04	0.85865	85903	85941	85979	86017	86055	86093	86131	86169	86206
1.05	0.86244	86281	86318	86356	86393	86430	86467	86504	86541	86578
1.06	0.86614	86651	86688	86724	86760	86797	86833	86869	86905	86941
1.07	0.86977	87013	87049	87085	87120	87156	87191	87227	87262	87297
1.08	0.87333	87368	87403	87438	87473	87507	87542	87577	87611	87646
1.09	0.87680	87715	87749	87783	87817	87851	87885	87919	87953	87987
1.10	0.88021	88054	88088	88121	88155	88188	88221	88254	88287	88320
1.11	0.88353	88386	88419	88452	88484	88517	88549	88582	88614	88647
1.12	0.88679	88711	88743	88775	88807	88839	88871	88902	88934	88966
1.13	0.88997	89029	89060	89091	89122	89154	89185	89216	89247	89277
1.14	0.89308	89339	89370	89400	89431	89461	89492	89552	89552	89582
1.15	0.89612	89642	89672	89702	89732	89762	89792	89821	89851	89880
1.16	0.89910	89939	89968	89997	90027	90056	90085	90114	90142	90171
1.17	0.90200	90229	90257	90286	90314	90343	90371	90399	90428	90456
1.18	0.90484	90512	90540	90568	90595	90623	90651	90678	90706	90733
1.19	0.90761	90788	90815	90843	90870	90897	90924	90951	90978	91005
1.20	0.91031	91058	91085	91111	91138	91164	91191	91217	91243	91269
1.21	0.91296	91322	91348	91374	91399	91425	91451	91477	91502	91528
1.22	0.91553	91579	91604	91630	91655	91680	91705	91730	91755	91780
1.23	0.91805	91830	91855	91879	91904	91929	91953	91978	92002	92026
1.24	0.92051	92075	92099	92123	92147	92171	92195	92219	92243	92266
1.25	0.92290	92314	92337	92361	92384	92408	92431	92454	92477	92500
1.26	0.92524	92547	92570	92593	92615	92638	92661	92684	92706	92729
1.27	0.92751	92774	92796	92819	92841	92863	92885	92907	92929	92951
1.28	0.92973	92995	93017	93039	93061	93082	93104	93126	93147	93168
1.29	0.93190	93211	93232	93254	93275	93296	93317	93338	93359	93380
1.30	0.93401	93422	93442	93463	93484	93504	93525	93545	93566	93586
1.31	0.93606	93627	93647	93667	93687	93707	93727	93747	93767	93787
1.32	0.93807	93826	93846	93866	93885	93905	93924	93944	93963	93982
1.33	0.94002	94021	94040	94059	94078	94097	94116	94135	94154	94173
1.34	0.94191	94210	94229	94247	94266	94284	94303	94321	94340	94358
1.35	0.94376	94394	94413	94431	94449	94467	94485	94503	94521	94538
1.36	0.94556	94574	94592	94609	94627	94644	94662	94679	94697	94714
1.37	0.94731	94748	94766	94783	94800	94817	94834	94851	94868	94885
1.38	0.94902	94918	94935	94952	94968	94985	95002	95018	95035	95051
1.39	0.95067	95084	95100	95116	95132	95148	95165	95181	95197	95213
1.40	0.95229	95244	95260	95276	95292	95307	95323	95339	95354	95370
1.41	0.95385	95401	95416	95431	95447	95462	95477	95492	95507	95523
1.42	0.95538	95553	95568	95582	95597	95612	95627	95642	95656	95671
1.43	0.95686	95700	95715	95729	95744	95758	95773	95787	95801	95815
1.44	0.95830	95844	95858	95872	95886	95900	95914	95928	95942	95956
1.45	0.95970	95983	95997	96011	96024	96038	96051	96063	96078	96092
1.46	0.96105	96119	96132	96145	96159	96172	96185	96198	96211	96224
1.47	0.96237	96250	96263	96276	96289	96302	96315	96327	96340	96353
1.48	0.96365	96378	96391	96403	96416	96428	96440	96453	96465	96478
1.49	0.96490	96502	96514	96526	96539	96551	96563	96575	96587	96599
x	0	2	4	6	8	x	0	2	4	6
1.50	0.96611	96634	96658	96681	96705	1.96	0.99443	99447	99452	99457
1.51	0.96728	96751	96774	96796	96819	1.97	0.99466	99471	99476	99480
1.52	0.96841	96864	96886	96908	96930	1.98	0.99489	99494	99498	99502
1.53	0.96952	96973	96995	97016	97037	1.99	0.99511	99515	99520	99524
1.54	0.97059	97080	97100	97121	97142	2.00	0.99532	99536	99540	99544
1.55	0.97162	97183	97203	97223	97243	2.01	0.99552	99556	99560	99564
1.56	0.97263	97283	97302	97322	97341	2.02	0.99572	99576	99580	99583
1.57	0.97360	97379	97398	97417	97436	2.03	0.99591	99594	99598	99601
1.58	0.97455	97473	97492	97510	97528	2.04	0.99609	99612	99616	99619
1.59	0.97546	97564	97582	97600	97617	2.05	0.99626	99629	99633	99636
1.60	0.97635	97652	97670	97687	97704	2.06	0.99642	99646	99649	99652
1.61	0.97721	97738	97754	97771	97787	2.07	0.99658	99661	99664	99667
1.62	0.97804	97820	97836	97852	97868	2.08	0.99673	99676	99679	99682
1.63	0.97884	97900	97916	97931	97947	2.09	0.99688	99691	99694	99697
1.64	0.97962	97977	97993	98008	98023	2.10	0.99702	99705	99707	99710
1.65	0.98038	98052	98067	98082	98096	2.11	0.99715	99718	99721	99723
1.66	0.98110	98125	98139	98153	98167	2.12	0.99728	99731	99733	99736
1.67	0.98181	98195	98209	98222	98236	2.13	0.99741	99743	99745	99748
1.68	0.98249	98263	98276	98289	98302	2.14	0.99753	99755	99757	99759
1.69	0.98315	98328	98341	98354	98366	2.15	0.99764	99766	99768	99770
1.70	0.98379	98392	98404	98416	98429	2.16	0.99775	99777	99779	99781
1.71	0.98441	98453	98465	98477	98489	2.17	0.99785	99787	99789	99791
1.72	0.98500	98512	98524	98535	98546	2.18	0.99795	99797	99799	99801
1.73	0.98558	98569	98580	98591	98602	2.19	0.99805	99806	99808	99810
1.74	0.98613	98624	98635	98646	98657	2.20	0.99814	99815	99817	99819
1.75	0.98667	98678	98688	98699	98709	2.21	0.99822	99824	99826	99827
1.76	0.98719	98729	98739	98749	98759	2.22	0.99831	99832	99834	99836
1.77	0.98769	98779	98789	98798	98808	2.23	0.99839	99840	99842	99843
1.78	0.98817	98827	98836	98846	98855	2.24	0.99846	99848	99849	99851
1.79	0.98864	98873	98882	98891	98900	2.25	0.99854	99855	99857	99858
1.80	0.98909	98918	98927	98935	98944	2.26	0.99861	99862	99863	99865
1.81	0.98952	98961	98969	98978	98986	2.27	0.99867	99869	99870	99871
1.82	0.98994	99003	99011	99019	99027	2.28	0.99874	99875	99876	99877
1.83	0.99035	99043	99050	99058	99066	2.29	0.99880	99881	99882	99883
1.84	0.99074	99081	99089	99096	99104	2.30	0.99886	99887	99888	99889
1.85	0.99111	99118	99126	99133	99140	2.31	0.99891	99892	99893	99894
1.86	0.99147	99154	99161	99168	99175	2.32	0.99897	99898	99899	99900
1.87	0.99182	99189	99196	99202	99209	2.33	0.99902	99903	99904	99905
1.88	0.99216	99222	99229	99235	99242	2.34	0.99906	99907	99908	99909
1.89	0.99248	99254	99261	99267	99273	2.35	0.99911	99912	99913	99914
1.90	0.99279	99285	99291	99297	99303	2.36	0.99915	99916	99917	99918
1.91	0.99309	99315	99321	99326	99332	2.37	0.99920	99920	99921	99922
1.92	0.99338	99343	99349	99355	99360	2.38	0.99924	99924	99925	99926
1.93	0.99366	99371	99376	99382	99387	2.39	0.99928	99928	99929	99930
1.94	0.99392	99397	99403	99408	99413	2.40	0.99931	99932	99933	99933
1.95	0.99418	99423	99428	99433	99438	2.41	0.99935	99935	99936	99937

附录 B 误差函数值表

(续)

x	0	2	4	6	8	x	0	2	4	6	8
2.42	0.99938	99939	99939	99940	99940	2.47	0.99952	99953	99953	99954	99954
2.43	0.99941	99942	99942	99943	99943	2.48	0.99955	99955	99956	99956	99957
2.44	0.99944	99945	99945	99946	99946	2.49	0.99957	99958	99958	99958	99959
2.45	0.99947	99947	99948	99949	99949	2.50	0.99959	99960	99960	99961	99961
2.46	0.99950	99950	99951	99951	99952
x	0	1	2	3	4	5	6	7	8	9
2.5	0.99959	99961	99963	99965	99967	99969	99971	99972	99974	99975
2.6	0.99976	99978	99979	99980	99981	99982	99983	99984	99985	99986
2.7	0.99987	99987	99988	99989	99989	99990	99991	99991	99992	99992
2.8	0.99992	99993	99993	99994	99994	99994	99995	99995	99995	99996
2.9	0.99996	99996	99996	99997	99997	99997	99997	99997	99997	99998
3.0	0.99998	99998	99998	99998	99998	99998	99998	99998	99998	99999

附录 B 误差函数值表

\(J_{n}(\beta)\)

n\β	0.5	1	2	3	4	6	8	10	12
0	0.9385	0.7652	0.2239	-0.2601	-0.3971	0.1506	0.1717	-0.2459	0.0477
1	0.2423	0.4401	0.5767	0.3391	-0.0660	-0.2767	0.2346	0.0435	-0.2234
2	0.0306	0.1149	0.3528	0.4861	0.3641	-0.2429	-0.1130	0.2546	-0.0849
3	0.0026	0.0196	0.1289	0.3091	0.4302	0.1148	-0.2911	0.0584	0.1951
4	0.0002	0.0025	0.0340	0.1320	0.2811	0.3576	-0.1054	-0.2196	0.1825
5	—	0.0002	0.0070	0.0430	0.1321	0.3621	0.1858	-0.2341	-0.0735
6		—	0.0012	0.0114	0.0491	0.2458	0.3376	-0.0145	-0.2437
7			0.0002	0.0025	0.0152	0.1296	0.3206	0.2167	-0.7103
8			—	0.0005	0.0040	0.0565	0.2235	0.3179	0.0451
9				0.0001	0.0009	0.0212	0.1263	0.2919	0.2304
10				—	0.0002	0.0070	0.0608	0.2075	0.3005
11					—	0.0020	0.0256	0.1231	0.2704
12						0.0005	0.0096	0.0634	0.1953
13						0.0001	0.0033	0.0290	0.1201
14						—	0.0010	0.0120	0.0650