Radar Range Resolution Comes from Bandwidth, Velocity Resolution from Time, and a One-Wrap Velocity Error Shifts Range One Cell ── with 1 GHz of bandwidth the range resolution is c/2B=0.149896 m, and 30 m/s wrapping to -8.935373 m/s shifts the corrected range by 1.000033 range resolutions ── the separator is whether the velocity has been mistaken by one wrap ── [Paper 802]

Abstract

A continuous-wave radar that sweeps its frequency measures a target's range and velocity with one device. What sets the range resolution, the velocity resolution and the velocity limit is laid out, and it is counted how range and velocity mix within one chirp and what happens to the range reading when the velocity exceeds its limit. No new theorem or law is claimed. Scope of this paper (scope note): No new theorem or law is claimed──the range resolution c/2B of a frequency-modulated continuous-wave radar, the round-trip Doppler effect 2v/lambda, reading velocity from the phase across chirps, and the wrapping of velocity readings are all standard (Skolnik's handbook, the textbook by Richards et al.). No circuits or noise are built──mixers, filters, sampling implementation and receiver noise are left out (the same line as Paper 335). A single target moving at constant velocity as a point──extended targets, acceleration and separation of several targets are not treated. The numbers are examples──a centre frequency of 77 GHz, 1 GHz bandwidth, 50-microsecond chirps, 256 samples per chirp and 128 chirps are values for comparison. Methods for resolving the wrap are not treated──procedures such as combining chirps of different lengths are not the subject. No relativistic correction is included──the target speed is taken as small enough against the speed of light, and only the first-order round-trip Doppler formula is used. No device is designed──neither weapons nor products are designed. Relation to earlier papers: Paper 335 showed that the round trip changes the exponent of received strength, and its scope note stated that noise and propagation were not treated──this paper stands beside that seat: here the same round trip attaches a 2 to both range and velocity. Paper 491 showed that the classical Doppler effect is two formulas and relativity their geometric mean──here, with target speeds small against light, the difference between the formulas does not appear and the first-order round-trip formula suffices. Paper 649 showed that separating two tidal constituents requires a record as long as their beat period──velocity resolution set by observation time is the same type; here a difference in velocity becomes a difference in how the phase turns from chirp to chirp. Paper 266 showed that the sampling theorem is correct but its premise is never met──the wrapping of velocity is the aliasing of sampling, arising from picking up the phase once per chirp. What is added is showing that the ratio of velocity limit to resolution is set only by half the number of chirps, 64.000000, confirming numerically that the range shift within one chirp can be corrected with the velocity, showing by formula that correcting with a wrapped velocity shifts the range by exactly c/2B and confirming it numerically as 1.000033 resolutions, and placing the separator on whether the velocity has been mistaken by one wrap. First, range resolution is set by bandwidth alone, and velocity resolution by observation time──range 0.149896 m, velocity 0.304173 m/s (Section 2). Second, the ratio of velocity limit to velocity resolution is set only by half the number of chirps──limit +/-19.467043 m/s, ratio 64.000000 (Section 2). Third, within one chirp, velocity turns into range──0.003850 m per m/s; a 5 m target at 10 m/s appears at 5.038500 m (Section 3). Fourth, reading velocity from the phase across chirps undoes the shift──reading 9.996326 m/s and correcting gives 5.000014 m (Section 3). Fifth, and this is the core. Mistaking the velocity by one wrap moves the range by exactly one cell──30 m/s wraps to -8.935373 m/s, the range corrected with it is 5.149901 m, off by 1.000033 range resolutions (Section 4). Sixth, the separator is whether the velocity has been mistaken by one wrap──"the range shift within one chirp can be corrected with the read velocity" is not decided true or false until one says whether the velocity has been mistaken by one wrap (Section 5). A radar's range resolution is set by bandwidth and its velocity resolution by observation time, and mistaking the velocity by one wrap moves the range by exactly one cell. With 77 GHz, 1 GHz bandwidth, 50-microsecond chirps and 128 chirps, the range resolution is 0.149896 m, the velocity resolution 0.304173 m/s and the limit +/-19.467043 m/s, and the ratio of limit to resolution stays at half the number of chirps, 64.000000──limit and resolution are bought one at the cost of the other. Within one chirp, 1 m/s turns into 0.003850 m of range, but correcting with the velocity read from the phase across chirps brings the range back to 5.000014 m. Yet 30 m/s, beyond the limit, wraps to -8.935373 m/s, and the range corrected with it, 5.149901 m, is off by 1.000033 range resolutions──because the range that one wrap of velocity, lambda/(2T), turns into is, by formula, c/2B itself. The separator is whether the velocity has been mistaken by one wrap. Placed among the earlier papers──the round trip of Paper 335, the Doppler formulas of Paper 491, the record length of Paper 649 and the aliasing of sampling of Paper 266 were assembled in one device and counted. To be honest──neither circuits, noise nor extended targets are treated, and no method for resolving the wrap is stated. On the making of this work: The ideas and content of this work stem from the author's own considerations. Assistance from an AI (a large language model) was used for structuring, English translation, and checking the algebra. Any remaining errors or misinterpretations are solely the author's. Feedback and corrections are sincerely appreciated. Keywords: frequency-modulated continuous-wave radar, range resolution, velocity resolution, Doppler effect, aliasing, observation time. ----- 周波数を掃引する連続波のレーダーは、的までの距離と的の速さを一つの装置で測る。距離の細かさ・速さの細かさ・速さの上限を何が決めるかを並べ、一回の掃引の中で距離と速さが混ざることと、速さの上限を超えたときに距離の読みに何が起きるかを数える。新しい定理も法則も主張しない。 本稿の射程(射程注記):新しい定理も法則も主張しない──周波数変調連続波レーダーの距離の細かさ c/2B、往復のドップラー効果の 2v/lambda、掃引をまたぐ位相から速さを読むこと、速さの読みが折り返すことは、いずれも標準的である(スコルニクの便覧、リチャーズらの教科書)。回路と雑音を作らない──混合器、フィルタ、標本化の実装、受信の雑音は入れない(論文335 と同じ線)。的は一つで、等速で動く点とする──広がった的、加速、複数の的の分離は扱わない。数は例である──中心周波数 77 GHz、帯域 1 GHz、掃引 50 マイクロ秒、一回の掃引 256 点、128 回は比べるための値である。折り返しを解く方法を扱わない──掃引の長さを変えて組み合わせるような手順は主題ではない。相対論の補正を入れない──的の速さは光速に比べて十分小さいとし、往復のドップラーは一次の式だけを使う。装置を設計しない──兵器も製品も設計しない。既刊との関係:論文335 は、往復すると受信の強さの冪が変わることを示し、射程注記で雑音と電波伝搬を扱わないとした──本稿はその席の隣に置く。ここでは同じ往復が、距離にも速さにも 2 を付ける。論文491 は、古典のドップラーが二つの式であり、相対論がその幾何平均であることを示した──的の速さが光速に比べて小さいここでは、式の違いは現れず、往復の一次の式で足りる。論文649 は、潮汐表で二つの分潮を分けるには、そのうなりの周期と同じだけの長さの記録が要ることを示した──速さの細かさが観測時間で決まるのは同じ型で、ここでは速さの差が掃引ごとの位相の回り方の差になる。論文266 は、標本化定理は正しいがその前提は決して満たされないことを示した──速さの折り返しは、掃引ごとに位相を一回ずつ拾うことで起きる標本化の折り返しである。加えたのは、速さの上限と細かさの比が掃引の回数の半分 64.000000 だけで決まることを示したこと、一回の掃引の距離のずれを速さで直せることを数値で確かめたこと、折り返した速さで直すと距離がちょうど c/2B ずれることを式で示し、数値で細かさの 1.000033 倍と確かめたこと、分離子を「速さを一周取り違えたかどうか」に置いたことである。 第一に、距離の細かさは帯域だけで、速さの細かさは観測時間で決まる──距離 0.149896 m、速さ 0.304173 m/s(第2節)。 第二に、速さの上限と細かさの比は、掃引の回数の半分だけで決まる──上限 +/-19.467043 m/s、比 64.000000(第2節)。 第三に、一回の掃引の中では、速さが距離に化ける──1 m/s あたり 0.003850 m、5 m の的が 10 m/s なら 5.038500 m に見える(第3節)。 第四に、掃引をまたぐ位相で速さを読めば、化けた分を戻せる──9.996326 m/s と読んで直すと 5.000014 m(第3節)。 第五に、これが本稿の芯である。速さを一周取り違えると、距離がちょうど一ます動く──30 m/s は -8.935373 m/s に折り返し、その速さで直した距離は 5.149901 m、ずれは距離の細かさの 1.000033 倍(第4節)。 第六に、分離子は、速さを一周取り違えたかどうか──「一回の掃引の距離のずれは、読んだ速さで直せる」は、速さを一周取り違えていないかを言うまで正しいかどうかが決まらない(第5節)。 レーダーの距離の細かさは帯域で、速さの細かさは観測時間で決まり、速さを一周取り違えると距離がちょうど一ます動く。77 GHz、帯域 1 GHz、掃引 50 マイクロ秒、128 回と置くと、距離の細かさは 0.149896 m、速さの細かさは 0.304173 m/s、上限は +/-19.467043 m/s で、上限と細かさの比は掃引の回数の半分 64.000000 のまま動かない──上限と細かさは、一方を買えば他方を払う。一回の掃引の中では速さ 1 m/s が 0.003850 m の距離に化けるが、掃引をまたぐ位相で読んだ速さで直せば 5.000014 m に戻る。ところが上限を超えた 30 m/s は -8.935373 m/s に折り返し、その速さで直した距離 5.149901 m は、距離の細かさの 1.000033 倍ずれる──速さの一周 lambda/(2T) が化ける距離は、式の上で c/2B そのものだからである。分離子は、速さを一周取り違えたかどうかである。既刊との位置──論文335 の往復、論文491 のドップラーの式、論文649 の記録の長さ、論文266 の標本化の折り返しを、一つの装置に組んで数えた。正直に言えば──回路も雑音も広がった的も扱っておらず、折り返しを解く方法も述べていない。 作成にあたって:本稿の着想と内容は、著者自身の考察に基づくものです。文章の構成整理や英訳、数式の確認には AI(大規模言語モデル)の助力を得ました。最終的な内容の解釈や誤りがあれば、それらはすべて著者の責に帰します。お気づきの点があれば、ご教示いただければ幸いです。 キーワード:周波数変調連続波レーダー、距離の細かさ、速さの細かさ、ドップラー効果、折り返し、観測時間。